Gemara

Rav Chama bar Chanina: "Atzei Shitim Omdim" - Karshei ha'Mishkan stand the way they grew (the top of the board was higher on the tree than the bottom).

Sukah 41b (Mishnah - R. Yosi): If the first day of Sukos was on Shabbos and someone took a Lulav into Reshus ha'Rabim he is exempt, for he had permission.

Abaye: He is exempt if he was not Yotzei (he was obsessed with the Mitzvah), but if he was Yotzei he is liable.

Question: The moment he picked it up he was Yotzei!

Answer #1 (Abaye): It was upside down.

Answer #2 (Rava): It was in a Keli.

45b - Chizkiyah: One is Yotzei Mitzvos only the way they grew - "Atzei Shitim Omdim".

Support (Beraisa): "Atzei Shitim Omdim" - they stand the way they grew.

Rosh Hashanah 27b - Beraisa: If one inverted a Shofar and blew it, he was not Yotzei.

Rav Papa: This is not only if he turned it inside out like a shirt. It is even if he widened the narrow end and narrowed the wide end.

Question: What is the reason?

Answer: This is like Rav Masnah taught, "V'Ha'avarta" - it must be Derech Ha'avarah (the way the animal carried it when it was alive).

Rishonim

Rif (Sukah 20a) and Rosh (Sukah 3:32) bring the Gemara in Sukah.

Korban Nesan'el (9): When shaking the Lulav in each direction, one must ensure that it is not horizontal. Rather, the bottom must be lower. People are not careful about this.

Rif (Rosh Hashanah 6b) and Rosh (3:4): If one blew an inverted Shofar, he was not Yotzei. This is not if he turned it inside out like a shirt; rather, he widened the narrow end and narrowed the wide end. He is not Yotzei because we require "V'Ha'avarta".

Rosh (ibid.): Some explain 'This is not if he turned it inside out...', for obviously then it is Pasul. Others explain 'This is not if he turned it inside out...', for obviously then it is Kosher. The Yerushalmi is like the first opinion. It also requires to blow from the narrow end - "Min ha'Metzar Korasi Kah".

Rambam (Hilchos Lulav 7:9): Once one picks up the four Minim, whether he picked them up at once or one after the other, he was Yotzei. This is only if he picked them up the way they grow.

Rambam (Hilchos Shofar 1:6): If one widened the narrow end and narrowed the wide end and blew it, he was not Yotzei.

Kesef Mishneh and Beis Yosef (OC 586 DH Hofchu): The Rambam's text in Rosh Hashanah said 'This is not if he turned it inside out like a shirt. Then he would be Yotzei, for the narrow end is in its place, this is Derech Ha'avarah! Rather, he widened the narrow end...'

R. Mano'ach: The wide end is down, attached to the animal, and the narrow end is up. The Torah 'should' have said 'V'Tokata'. Rather, it says "V'Ha'avarta" to require Derech Ha'avarah. It is not clear what the Rambam holds regarding if it was turned inside out like a shirt. It appears that he holds that all the more so it is Pasul, like the Yerushalmi and accurate texts of the Rif. It is proper to be stringent.

Poskim

Rema (OC 651:9): When shaking the Lulav, one tilts the end of the Lulav in the direction in which he shakes. When shaking down, he inverts it. This is called the way it grows, since he holds it in his hands the way it grows.

Mishnah Berurah (45): This means that it suffices that he picked it up the way it grows.

Rema (ibid.): Some are meticulous not to invert it when shaking down. The custom is like the first opinion.

Beis Yosef (DH v'Rabeinu): Some texts of the Tur say that one does not tilt the top of the Lulav when shaking westward. Since it is in back of himself, it is clear that he shakes in that direction. Alternatively, one never tilts the Lulav; it is always shaken vertical, the way it grows. The custom is to tilt the end of the Lulav in every direction in which he shakes, including downwards. My father says that just like one is Yotzei the Mitzvah only the way they grow, the same applies to the shakings.

Question (Darchei Moshe 9): The Maharil would tilt the Lulav in every direction, except for down. If we are concerned for the way it grows, it should be held vertical when shaking in every direction!

Answer (Darchei Moshe 9): Regarding the other directions it is merely tilted slightly, this is still considered the way it grows. If one would point it down, this is not the way it grows.

Gra (DH u'Matin): The first opinion in the Rema learned from the Darchei Moshe's question that we are not concerned for the way it grows while shaking.

Taz (12): Shtei ha'Lechem are put on top of Kisvei Atzeres. The Kohen puts his hands underneath and waves in every direction (Menachos 61a). The loaves are on top even when he waves down. The Lulav is waved the same way (Sukah 38b). Also, if the Lulav points down when shaking down, one would have to tilt it to the side first to reach this position; the Gemara does not mention this. Further, if it points down one is not Yotzei according to the opinion that requires the way it grows.

Mishnah Berurah (46): It is best not to invert it, like the Taz.

Kaf ha'Chayim (99,100): The Ari Zal says that it should be held vertical for all the shakings. Also, it is disgraceful to hold it upside down.

Shulchan Aruch (586:12): If one inverted a Shofar and blew it, whether he turned it inside out like a shirt or widened the narrow end and narrowed the wide end, he was not Yotzei.

Mishnah Berurah (61): He widened and narrowed it by heating it in boiling water. It is Pasul because it is not the way it grew; the short side must be by the man's mouth.

Kaf ha'Chayim (104): Even if he widened the narrow end or narrowed the wide end, this is not Derech Ha'avarah, so he is not Yotzei.

Rema: Likewise, if he blew through the narrow end he was not Yotzei.

Mishnah Berurah (62): Also this is not the way it grew.