  A novel lemma in $\mathbb{Z}_2$-equivariant homotopy theory is stated, proven
and applied to the topological classification of frustrated magnets in the
presence of canonical time-reversal symmetry. This lemma generalises a result
which had been key to the homotopical derivation of the renowned Bott-Kitaev
periodic table for topological insulators and superconductors. We distinguish
between three symmetry classes $\mathrm{AIII}$, $\mathrm{AIII/BDI}$, and
$\mathrm{AIII/CII}$ depending on the existence and type of canonical
time-reversal symmetry. For each of these classes, the relevant objects to
classify are $\mathbb{Z}_2$-equivariant maps into a Stiefel manifold. The
topological classification is illustrated through examples of frustrated spin
models and is compared to the one of Roychowdhury and Lawler (RL).
