Gemara

R. Yehudah says that one must be able to hold the Esrog in one hand, lest he drop it and it become disqualified.

Sometimes one takes the Minim in the wrong hands, and must switch them;

(Rabah): One must hold the Lulav [bundle] in the right hand, and the Esrog in the left.

37b: This is because the Lulav [bundle] has three Mitzvos, and the Esrog is only one.

Shabbos 103a: A lefty's left hand is like a regular person's right hand. He is liable for writing with his left hand, and exempt for his right.

Pesachim 108a: Lying on one's back is not considered reclining.

Reclining on the right side is not considered reclining.

Further, it is dangerous, lest the food enter the windpipe instead of the esophagus.

Rishonim

Rosh (3:25): A lefty takes the Lulav in the left hand, for it is his Yemin (strong hand).

Rosh (Pesachim 10:20): Reclining on the right side is not called reclining. The Rashbam says that also there is concern lest food go down the windpipe. Therefore, a lefty reclines on his right side. Regarding concern lest food go down the windpipe, he is like others.

Terumas ha'Deshen (136): Some Ge'onim say that concern lest food go down the windpipe is only when one reclines on his back. Others say that it applies also to reclining on the right. The Rosh did not decide. We neglect the reason of what is called reclining, for danger is more severe than Isur, like it says in Chulin (10a).

Poskim

Shulchan Aruch (OC 27:2): The proper custom is that the Yud (the protrusion of the end of the strap next to the box) is towards the heart, and the Tefilah (the box that houses the parchment) faces out [from the body].

Shulchan Aruch (3): The proper custom is that the Tefilah is closer to the elbow, and the Ma'avarta (where the strap goes through) is closer to the shoulder.

Bi'ur Halachah (DH ha'Minhag): Shevus Yakov says that if a righty has only a lefty's Tefilin or vice-versa, he can reverse it, so the box is closer to the shoulder, and the Ma'avarta is closer to the elbow. Artzos ha'Chayim says that this is better than putting it on normally, and the Yud will face out. The Gra (Sa'if 2) says that the Ge'onim hold that the Gemara mandates this, so this overrides the custom that the Shulchan Aruch brings [in this Sa'if].

Shulchan Aruch (6): If a lefty does all tasks with his left hand, he puts Tefilin on the right hand. If one writes with his left hand, but does all other tasks with his right hand, or vice-versa, Poskim argue about which hand he puts Tefilin on.

Kaf ha'Chayim (31): If one does all tasks with his left hand, he puts Tefilin on his left hand and blesses, and afterwards puts Tefilin on his right hand. It is best if he has another Tefilin Shel Yad, so on each arm the Ma'avarta can face his shoulder. If not, he borrows from a righty, or puts his Tefilin on with the Ma'avarta towards the elbow, so the Yud will be towards the heart.

Shulchan Aruch (183:5): Some say that if the one saying Birkas ha'Mazon aloud is a lefty, he holds the cup in his left hand.

Source (Beis Yosef DH u'Mah): Shibolei ha'Leket says that R. Binyamin learns this from Tefilin.

Magen Avraham (9): This is difficult. In Siman 651 the Beis Yosef says since which hand we use to take the Lulav is only mid'Rabanan, it depends on most people! All the more so here it should depend on most people, for the cup of Birkas ha'Mazon is totally mid'Rabanan! Our texts of Shibolei ha'Leket do not mention this. (Note: Our text of Shibolei ha'Leket (156) says "he puts it in his Yemin, which is his Smol" - PF.) He said to take the Lulav with the right hand. Perhaps another Posek said so about Birkas ha'Mazon. In any case, I say that one should do so (take it in his strong hand), like the Rema says about Lulav.

Mishnah Berurah (20): All these matters (e.g. in which hand we hold the cup) are only Hidur Mitzvah l'Chatchilah.

Kaf ha'Chayim (29): The Mechaber said 'some say that' because this is not primary. Regarding Lulav, in 651:3 he brings from Ba'al ha'Itur and the Tur that a lefty is like others. Also here, the law is mid'Rabanan. According to Kabalah, all must hold the cup in the right hand. The Kabalah equates a lefty to others even regarding Tefilin!

Rema (472:3): At the Seder, also a lefty leans on his left side.

Mishnah Berurah (10): Even though a lefty eats with his left hand, the danger overrides. He reclines on his left, and eats with his right hand. B'Di'eved, if he reclined on his right, he was Yotzei, for this is like others' left. (It is the convenient way to eat.)

Bach (3 DH v'Chasav, citing Maharshal): The Gemara said that reclining on the right is not called reclining. If one did, he was not Yotzei. However, according to those who forbid due to danger, he already endangered himself, and it will not help to eat again reclining on the left! However, we hold like Avi ha'Ezri, who says that he was not Yotzei b'Di'eved even without reclining.

Rejection (Pri Chodosh 3): Since it is not called reclining, he did not fulfill the enactment. He must eat again, reclining.

Kaf ha'Chayim (23): The Pri Chodosh connotes that b'Di'eved he was not Yotzei.

Note: He did not distinguish lefties from righties. According to the Terumas ha'Deshen and Rema, he was obligated to recline on the left, and he did not fulfill this.

Bi'ur Halachah (DH v'Ein): If one's right hand is maimed and he cannot eat with it, what should he do? It is dangerous to lean to the right. If he leans to the left, he cannot eat! Perhaps in such a case, Chachamim did not obligate him to lean. This requires investigation.

Kaf ha'Chayim (22): Chachamim enacted reclining in the way of freedom and pleasure, but not for pain. Likewise, if one cannot recline on his left due to a wound there, he is exempt from reclining. We rely on Avi ha'Ezri, brought in Rema Sa'if 4 [that nowadays we do not recline].

Shulchan Aruch (651:3): A lefty takes the Lulav in his right hand and the Esrog in his left. It depends others' right and left, and not on his.

Beis Yosef (DH u'Mah): He takes the Esrog in the right, for it is his Smol (weak hand). It depends on Yemin and Smol for each person. The Tur's words "he takes the Esrog in what is Smol for others" connote oppositely, that he takes in the right, i.e. his weak hand, for it depends on most people. R. Yerucham brings from Tosfos and Maharam that one who always uses his left hand takes the Lulav in the left, which is his Yemin. This is like the Rosh. Ba'al ha'Itur says that presumably, for Tefilin, which is mid'Oraisa, we require Yemin. Lulav is mid'Rabanan, for importance. A lefty is like everyone, and takes it in his right, like all others. If he did oppositely, there is no problem. It is no worse than Tefilin. (I.e. he uses his left hand [to tie Tefilin on his weak hand], and he is Yotzei - PF.) The Tur rules like Ba'al ha'Itur. He says that Lulav is mid'Rabanan, i.e. the Mitzvah to hold it in the right hand.

Taz (3): The Beis Yosef says that most Poskim hold that it depends on his own right and left. The Tur himself wrote so in Kitzur Piskei ha'Rosh, unlike he wrote in the Tur. Why didn't he even mention the Rosh's opinion, like he always does? Also, if he takes it in his weak hand, how can he shake it? Surely, there is a printing mistake in the Tur. It should say 'he takes the bundle in what is others' left hand... and even if he is a lefty.' I.e. even then it depends on his strong and weak hand. Also the Maharshal corrected the text. Surely, we follow the Rema, who rules like the Rosh and other Poskim. Why did the Levush say that this is not the custom? The Rema said that it is the custom! Do not heed one who does oppositely.

Rema: Some say that it depends on his right and left. He takes the Lulav in his left and the Esrog in his right. This is the custom. It is primary. If he did oppositely, he was Yotzei.

Taz (4): Why did the Rema write this? He connotes that if a righty switched, he was not Yotzei. This is wrong. It is only l'Chatchilah to take in the right hand the bundle with three Mitzvos. The Torah did not command in which hand to take them. This is clear from 42a. We asked that once he picked them up, he was Yotzei. Abaye answered that he picked them up upside-down. He did not say that he took the Lulav in his left hand! The Poskim say to hold them upside-down before the Berachah. They do not say to take the Lulav in the left hand, for in any case he was Yotzei!

Note: Perhaps the Poskim did not suggest this, for changing hands could cause one to drop them and disqualify them, like it says on 31b!

Magen Avraham (9): The Rema connotes that even if a righty reversed them, he was Yotzei. On 42a, we say that if he switched [he was not Yotzei]. I saw in the name of R. Chananel that he took the Lulav in the left, and the Esrog in the right. He holds that one is not Yotzei this way. [If one did so] he should take them again, without a Berachah.

Gra (DH v'Im): The Rema means that even if a righty reversed them, he was Yotzei, unlike some who explain 42a to say oppositely (Ritva 42a DH Gemara).

Rema: An ambidextrous person takes the Lulav in the right, like most people.

Mishnah Berurah (20): This is when he does all tasks equally with either hand. If he can use his right, but it is easier with his left, this is not called ambidextrous.

Kaf ha'Chayim (41): If it is easier with his left, but he writes with his right hand, he takes the Lulav in his right hand.

Kaf ha'Chayim (38): Ma'amar Mordechai says that it is proper to rule like the Rema and other Acharonim, but the custom is like the Mechaber, i.e. the Sefardi custom. After blessing and taking like the Mechaber says, one should switch hands to fulfill the other opinion. Likewise, one who follows the Rema should bless, take them, and switched hands to fulfill all opinions, since there is no loss in this. One who is stringent will be blessed.

Kaf ha'Chayim (39): The Rambam (Hilchos Lulav 7:19) clearly says that even if a righty switched, he was Yotzei.

Kaf ha'Chayim (40): If one writes with his right hand, but does other tasks with his left he takes the Lulav in the left. Writing is significant only for Tefilin. "U'Kshartam...; U'Chsavtam" teaches that like we use the Yemin to write, we use it to tie. However, afterwards he should switch hands.