  In the electroweak sector of the Standard Model, CP violation arises through
a very particular interplay between the three quark generations, as described
by the Cabibbo--Kobayashi--Maskawa (CKM) mechanism and the single Jarlskog
invariant $J_4$. Once generalized to the Standard Model Effective Field Theory
(SMEFT), this peculiar pattern gets modified by higher-dimensional operators,
whose associated Wilson coefficients are usually split into CP-even and odd
parts. However, CP violation at dimension four, i.e., at the lowest order in
the EFT expansion, blurs this distinction: any Wilson coefficient can interfere
with $J_4$ and mediate CP violation. In this paper, we study such interferences
at first order in the SMEFT expansion, $\mathcal{O}(1/\Lambda^2)$, and we
capture their associated parameter space via a set of 1551 linear CP-odd flavor
invariants. This construction describes both new, genuinely CP-violating
quantities as well as the interference between $J_4$ and CP-conserving ones. We
call this latter possibility \textit{opportunistic CP violation}. Relying on an
appropriate extension of the matrix rank to Taylor expansions, which we dub
\emph{Taylor rank}, we define a procedure to organize the invariants in terms
of their magnitude, so as to retain only the relevant ones at a given
precision. We explore how this characterization changes when different
assumptions are made on the flavor structure of the SMEFT coefficients.
Interestingly, some of the CP-odd invariants turn out to be less suppressed
than $J_4$, even when they capture opportunistic CPV, demonstrating that
CP-violation in the SM, at dimension 4, is \textit{accidentally small.}
