In my game I tried adding rotation to one of the objects, but this led to a problem: Before all my textured rectangles were at the origin, with the corner at (0,0). But to rotate them I realized they would need to be centered around the origin. But all my game rectangles use the upper-left corner of the rectangle as their origin, so translating the matrix by the game rectangle's coordinate results in the center being at where the corner should be. I guess I could just translate by rect.origin.x + rect.size.width / 2, but... I don't know this doesn't feel as clean as it was before somehow. Is there any other solution to this?

If you are using textured quads, it is probably best to render them centered to facilitate rotation. This should not be hard to do, just use square verts, e.g.:
static const GLfloat squareVertices[] = {
-SZ, -SZ,
SZ, -SZ,
-SZ, SZ,
SZ, SZ,
};

Also remember that typically you apply both a translation and a rotation to your game objects - and the order matters.
eg.
glTranslate followed by glRotate results differently than glRotate followed by glTranslate.

Hope that helps...

Yea that's how I was doing it. I think the thing I didn't like was how I was working with matrices. I'm using legacy-free OpenGL code so no glTranslate and friends. I'm writing my own, with a 2d game in mind, but I don't think I have it right. OpenGL always multiplies the matrix for even glTranslate, but isn't 64 multiplications and 48 additions a waste when usually all you need is two additions? Also some of my game objects only need a rectangle, others need a rectangle and a velocity, others need that plus a rotation, so I'm not sure if I should make those combinations all different structs. But there's no point in calculating rotation for something that never rotates.