  Solid state systems with time reversal symmetry and/or particle-hole symmetry
often only have $\mathbb{Z}_2$-valued strong invariants for which no general
local formula is known. For physically relevant values of the parameters, there
may exist approximate symmetries or almost conserved observables, such as the
spin in a quantum spin Hall system with small Rashba coupling. It is shown in a
general setting how this allows to define robust integer-valued strong
invariants stemming from the complex theory, such as the spin Chern numbers,
which modulo $2$ are equal to the $\mathbb{Z}_2$-invariants. Moreover, these
integer invariants can be computed using twisted versions of the spectral
localizer.
