  We produce a class of $\omega$-categorical structures with finite signature
by applying a model-theoretic construction -- a refinement of the
Hrushosvki-encoding -- to $\omega$-categorical structures in a possibly
infinite signature. We show that the encoded structures retain desirable
algebraic properties of the original structures, but that the constraint
satisfaction problems (CSPs) associated with these structures can be badly
behaved in terms of computational complexity. This method allows us to
systematically generate $\omega$-categorical templates whose CSPs are complete
for a variety of complexity classes of arbitrarily high complexity, and
$\omega$-categorical templates that show that membership in any given
complexity class cannot be expressed by a set of identities on the
polymorphisms. It moreover enables us to prove that recent results about the
relevance of topology on polymorphism clones of $\omega$-categorical structures
also apply for CSP templates, i.e., structures in a finite language. Finally,
we obtain a concrete algebraic criterion which could constitute a description
of the delineation between tractability and NP-hardness in the dichotomy
conjecture for first-order reducts of finitely bounded homogeneous structures.
