Picture: Somebody holding a Sephardi‐style Torah (left) and somebody holding an Ashkenazi‐style Torah (right).

One of the many fascinating differences between Ashkenazim and Sephardim is that the former fit their Torahs on two rollers, which they usually wrap together in a cloth before putting away, whereas the latter have a more elaborate method: storing the Torah in a case (traditionally made from wood) with built‐in rollers, and reading it therefrom!

The reason for this divergence is actually more complicated than you might have expected. Quoting Rabbi Yehuda Shurpin:

On the surface, one might think that this difference is primarily a cultural one. Just like people in different lands dress differently, they also developed different styles for “dressing” the Torah.

However, there actually is important halachic significance to the fact that the Sephardim keep their Torahs in cylinders that hold them upright, while the Ashkenazi Torah scrolls need to be propped up on a slant.

[…]

The Talmud states that if one positions the mezuzah like a nagar (bolt), it is invalid. Now, what is the position of a bolt?

The great commentator Rashi (Rabbi Shlomo Yitzchaki) explains that a nagar is a bolt that lies horizontally. Thus, he learns that a mezuzah needs to be attached vertically.¹

Rashi’s grandson, Rabbeinu Tam (Rabbi Yaakov ben Meir), disagrees, saying that the nagar in question is not a horizontal door bolt, but a vertical tent pin, like the pins used to keep the Tabernacle walls in place. Accordingly, he says that a vertical mezuzah is invalid. Interestingly, he finds a parallel for his horizontal requirement in the fact that the dead must be buried lying down, and in the historical precedent that the Tablets were positioned on their sides in the Holy Ark.²

Now, if the Tablets and mezuzahs are not allowed to stand upright, it follows that Rabbeinu Tam would also forbid having a Torah stand vertically.

What This Means to Us

Maimonides³ and the Code of Jewish Law⁴ concur with Rashi’s opinion that the correct position of a mezuzah is in the vertical position, and if it’s in the horizontal position, it is invalid.

Thus, Sephardi[m] generally affix their mezuzot in the vertical position, and extend this to the Torah scrolls, which they store and read while the scrolls are standing in their cases.

Ashkenazic custom, however, generally follows the opinion of the Rema (Rabbi Moses Isserles) in his glosses to the Code of Jewish Law. He writes that although the halachah follows Rashi’s opinion, one should take Rabbeinu Tam’s opinion (that a vertical mezuzah is invalid) into consideration. Therefore, he suggests that the best way to affix the mezuzah is diagonally (with the top facing inward). Since the mezuzah is neither vertical nor horizontal, it is valid according to both opinions.⁵

And now you can also see why Ashkenazim read the Torah on a slanted bimah and are particular to store the scrolls on a (slight) slant as well.⁶

(Source.)

Although it’s somewhat unrelated, I do wish that the Ashkenazic questioner wrote more about experiencing a Sephardic synagogue for the first time. The simultaneous feelings of familiarity and unfamiliarity must have been quite interesting indeed.