Several authors have tried to give equations describing the curve traced out on the surface of a baseball by its seam. Taking an alternative approach, we identify several symmetry properties that any reasonable candidate for a baseball curve must have and call the class of curves on the sphere that have these symmetries generalized baseball curves. We show that curves whose geodesic curvature functions satisfy certain symmetries are in fact generalized baseball curves as long as they are closed or periodic. We look at the family of curves whose geodesic curvature is of the form sin(cs), and show that there are many such curves that do close back on themselves and are therefore generalized baseball curves. Along the way, we come up with what we think is a nice and previously undiscussed curve that fits the actual seam of a baseball surprisingly well.