  There is a long standing challenge in lattice QCD concerning the relationship
between $\mathcal{CP}$-symmetry and lattice chiral symmetry: na\"ively the
chiral symmetry transformations are not invariant under $\mathcal{CP}$. With
results similar to a recent work by Igarashi and Pawlowski, I show that this is
because charge conjugation symmetry has been incorrectly realised on the
lattice. The naive approach, to directly use the continuum charge conjugation
relations on the lattice, fails because the renormalisation group blockings
required to construct a doubler free lattice theory from the continuum are not
invariant under charge conjugation. Correctly taking into account the
transformation of these blockings leads to a modified lattice $\mathcal{CP}$
symmetry for the fermion fields, which, for gauge field configurations with
trivial topology, has a smooth limit to continuum $\mathcal{CP}$ as the lattice
spacing tends to zero. After constructing $\mathcal{CP}$ transformations for
one particular group of lattice chiral symmetries, I construct a lattice chiral
gauge theory which is $\mathcal{CP}$ invariant and whose measure is invariant
under gauge transformations and $\mathcal{CP}$.
