  A fundamental dichotomous classification for all physical systems is
according to whether they are spinless or spinful. This is especially crucial
for the study of symmetry-protected topological phases, as the two classes have
distinct symmetry algebra. As a prominent example, the spacetime inversion
symmetry $PT$ satisfies $(PT)^2=\pm 1$ for spinless/spinful systems, and each
class features unique topological phases. Here, we reveal a possibility to
switch the two fundamental classes via $\mathbb{Z}_2$ projective
representations. For $PT$ symmetry, this occurs when $P$ inverses the gauge
transformation needed to recover the original $\mathbb{Z}_2$ gauge connections
under $P$. As a result, we can achieve topological phases originally unique for
spinful systems in a spinless system, and vice versa. We explicitly demonstrate
the claimed mechanism with several concrete models, such as Kramers degenerate
bands and Kramers Majorana boundary modes in spinless systems, and real
topological phases in spinful systems. Possible experimental realization of
these models is discussed. Our work breaks a fundamental limitation on
topological phases and opens an unprecedented possibility to realize intriguing
topological phases in previously impossible systems.
