Motivated by the study of periodic Hamiltonians enjoying chiral or particle-hole symmetry, like the SSH model or the Kitaev chain, we present a topological study of families of symmetric functions from d-dimensional tori to the space of fixed-rank projectors on a Hilbert space. Our approach is based on the direct calculation of fundamental groups and homotopy groups on spaces of unitary matrices, building on the existing classification of vector bundles in reduced dimensions. The main results include a classification up to unitary equivalences or homotopies for d = 0, 1 or 2 for chiral or particle-hole symmetric projectors within separable Hilbert spaces of minimal or infinite dimension.