<p>We present two broad sets of functionals for which the phase-space Feynman path integrals of the Schrödinger type on the torus are rigorously formulated. More precisely, for any functional in the union of the two sets, the time-slicing approximation via pairs of piecewise constant position and momentum paths converges uniformly on compact sets of terminal positions and initial momenta. Each set is closed under addition, multiplication, path translations, linear transformations of paths by integer matrices, and functional differentiation, enabling the construction of various path-integrable functionals. Furthermore, although some care is required, these phase-space path integrals satisfy several fundamental properties analogous to those of classical integrals: interchange of path integration with time integration and limit operations, invariance under path translations and orthogonal integer transformations of paths, and integration by parts for functional differentiation of position paths.</p>