  We study a homoclinic flip bifurcation of case~\textbf{C}, where a homoclinic
orbit to a saddle equilibrium with real eigenvalues changes from being
orientable to nonorientable. This bifurcation is of codimension two, and it is
the lowest codimension for a homoclinic bifurcation of a real saddle to
generate chaotic behavior in the form of (suspended) Smale horseshoes and
strange attractors. We present a detailed numerical case study of how global
stable and unstable manifolds of the saddle equilibrium and of bifurcating
periodic orbits interact close to such bifurcation. This is a step forward in
understanding the generic cases of homoclinic flip bifurcations, which started
with the study of the simpler cases \textbf{A} and \textbf{B}. In a
three-dimensional vector field due to Sandstede, we compute relevant
bifurcation curves in the two-parameter bifurcation diagram near the central
codimension-two bifurcation in unprecedented detail. We present representative
images of invariant manifolds, computed with a boundary value problem setup,
both in phase space and as intersection sets with a suitable sphere. In this
way, we are able to identify infinitely many cascades of homoclinic
bifurcations that accumulate on specific codimension-one heteroclinic
bifurcations between an equilibrium and various saddle periodic orbits. Our
computations confirm what is known from theory but also show the existence of
bifurcation phenomena that were not considered before. Specifically, we
identify the boundaries of the Smale--horseshoe region in the parameter plane,
one of which creates a strange attractor that resembles the R\"{o}ssler
attractor. The computation of a winding number reveals a complicated overall
bifurcation structure in the wider parameter plane that involves infinitely
many further homoclinic flip bifurcations associated with so-called homoclinic
bubbles.
