  The violation of charge conjugation (C) and parity (P) symmetries are a
requirement for the observed dominance of matter over antimatter in the
Universe. As an established effect of beyond the Standard Model physics, this
could point towards additional CP violation in the Higgs-gauge sector. The
phenomenological footprint of the associated anomalous couplings can be small,
and designing measurement strategies with the highest sensitivity is therefore
of the utmost importance in order to maximise the discovery potential of the
Large Hadron Collider (LHC). There are, however, very few measurements of
CP-sensitive observables in processes that probe the weak-boson
self-interactions. In this article, we study the sensitivity to new sources of
CP violation for a range of experimentally-accessible electroweak processes,
including $W\gamma$ production, $WW$ production via photon fusion, electroweak
$Zjj$ production, electroweak $ZZjj$ production, and electroweak $W^\pm W^\pm
jj$ production. We study simple angular observables as well CP-sensitive
observables constructed using the outputs of machine-learning (ML) algorithms.
We find that the ML-constructed CP-sensitive observables improve the
sensitivity to CP-violating effects by up to a factor of five, depending on the
process. We also find that inclusive $W\gamma$ and electroweak $Zjj$ production
have the potential to set the best possible constraints on certain CP-odd
operators in the Higgs-gauge sector of dimension-six effective field theories.
