  The possibility of selecting magnetic space groups by orienting the
magnetization direction or tuning magnetic orders offers a vast playground for
engineering symmetry protected topological phases in magnetic materials. In
this work, we study how selective tuning of symmetry and magnetism can
influence and control the resulting topology in a 2D magnetic system, and
illustrate such procedure in the ferromagnetic monolayer MnPSe$_3$. Density
functional theory calculations reveals a symmetry-protected accidental
semimetalic (SM) phase for out-of-plane magnetization which becomes an
insulator when the magnetization is tilted in-plane, reaching band gap values
close to $100$ meV. We identify an order-two composite antiunitary symmetry and
threefold rotational symmetry that induce the band crossing and classify the
possible topological phases using symmetry analysis, which we support with
tight-binding and $\mathbf{k}\cdot\mathbf{p}$ models. Breaking of inversion
symmetry opens a gap in the SM phase, giving rise to a Chern insulator. We
demonstrate this explicitly in the isostructural Janus compound
Mn$_2$P$_2$S$_3$Se$_3$, which naturally exhibits Rashba spin-orbit coupling
that breaks inversion symmetry. Our results map out the phase space of
topological properties of ferromagnetic transition metal phosphorus
trichalcogenides and demonstrate the potential of the magnetization-dependent
metal-to-insulator transition as a spin switch in integrated two-dimensional
electronics.
