Mishnah Berurah — משנה ברורה

Inside out - And people will see the ugly stitching and fasteners of the shirt and it will be disgraceful in the eyes of the people. And if one was not careful and did not reverse it - if he is a Torah scholar, he must remove it and put it back on the right way so that he will not be in the category of those who, Heaven forbid, cause others to hate Hashem; and for all other people, it is not necessary. But for prayers, even all other people must remove it and put it on the right way, for it is proper to wear honorable clothing, as is explained in siman 91. It is proper that one should hold both ends of the garment in his right hand and then put on the right side first and then the left side and he should have in mind that everything is included in the right [side] and from right it comes to the left. The Rambam wrote that the clothing of a Torah scholar must be pleasant and clean and it's forbidden for there to be a stain or oily residue or a similar thing [on it]; and he shouldn't wear clothing of royalty that [will cause] everyone will stare at him, and not poor man's clothing which is degrading for the one wearing it, rather he should wear regular clothing which are pleasant. See there (in the Rambam) further.