Topology of Wannierizable Insulators

We study the freedom in choosing Wannier centers and the associated symmetry representations in noninteracting Wannierizable insulators. We compare Wyckoff deformations of atomic configurations with symmetry-preserving adiabatic deformations of occupied states, both with and without adding common occupied atomic orbitals. Explicit examples show that stable Wyckoff equivalence and adiabatic equivalence at fixed filling do not imply each other. In spinless $Fm'm'm'$, an Euler class distinguishes stably Wyckoff-equivalent configurations. In spinless $P\overline1'$, adiabatically equivalent configurations remain distinct under stable Wyckoff deformations. We also classify when stable Wyckoff and stable adiabatic equivalence coincide. They agree in two or fewer dimensions and in three-dimensional spinful systems. For three-dimensional spinless systems, Chern insulator pumps give additional equivalences in 134 magnetic space groups. These groups preserve spacetime orientation and have a point whose site-symmetry group contains $222$, $\overline1'$, or $\overline4'$.