  The violation of CP symmetry at the electro-weak scale is one of the
essential ingredients for electro-weak baryogenesis. It is therefore of great
interest to map the CP properties of the Higgs sector in as much detail as
possible. A linear electron-positron Higgs factory collider will provide many
opportunities to probe the CP nature of the Higgs sector, thanks to access to
several Higgs production processes at a wide range of centre-of-mass energies.
In this paper we report on two studies based on full simulation of detector
response and realistic reconstruction algorithms: 1) a study of $h \to \tau
\tau$ at ILC-250, in which mixing of CP eigenstates can be measured to a
precision of 75~mrad; and 2) the current status of an ongoing study of the
$ZZ$--fusion process at 1.4 TeV CLIC and 1 TeV ILC, for which we expect to
achieve concrete results during the Snowmass study period.
