  Cobimaximal lepton mixing, i.e. $\theta_{23} = 45^\circ$ and $\delta = \pm
90^\circ$ in the lepton mixing matrix $V$, arises as a consequence of $S V =
V^\ast \mathcal{P}$, where $S$ is the permutation matrix that interchanges the
second and third rows of $V$ and $\mathcal{P}$ is a diagonal matrix of phase
factors. We prove that any such $V$ may be written in the form $V = U R P$,
where $U$ is any predefined unitary matrix satisfying $S U = U^\ast$, $R$ is an
orthogonal, i.e. real, matrix, and $P$ is a diagonal matrix satisfying $P^2 =
\mathcal{P}$. Using this theorem, we demonstrate the equivalence of two ways of
constructing models for cobimaximal mixing---one way that uses a standard $CP$
symmetry and a different way that uses a $CP$ symmetry including $\mu$--$\tau$
interchange. We also present two simple seesaw models to illustrate this
equivalence; those models have, in addition to the $CP$ symmetry, flavour
symmetries broken softly by the Majorana mass terms of the right-handed
neutrino singlets. Since each of the two models needs four scalar doublets, we
investigate how to accommodate the Standard Model Higgs particle in them.
