  We consider the model of a dual-core spatial-domain coupler with chi^(2) and
chi^(3) nonlinearities acting in two parallel cores. We construct families of
symmetric and asymmetric solitons in the system with self-defocusing chi^(3)
terms, and test their stability. The transition from symmetric to asymmetric
soliton branches, and back to the symmetric ones proceeds via a bifurcation
loop. A pair of stable asymmetric branches emerge from the symmetric family via
a supercritical bifurcation; eventually, the asymmetric branches merge back
into the symmetric one through a reverse bifurcation. The existence of the loop
is explained by means of an extended version of the cascading approximation for
the chi^(2) interaction, which takes into regard the XPM part of the chi(3)
interaction. When the inter-core coupling is weak, the bifurcation loop
features a concave shape, with the asymmetric branches losing their stability
at the turning points. In addition to the two-color solitons, which are built
of the fundamental-frequency (FF) and second-harmonic (SH) components, in the
case of the self-focusing chi^(3) nonlinearity we also consider single-color
solitons, which contain only the SH component, but may be subject to the
instability against FF perturbations. Asymmetric single-color solitons are
always unstable, whereas the symmetric ones are stable, provided that they do
not coexist with two-color counterparts. Collisions between tilted solitons are
studied too.
