  We study the nonstandard $q$-deformation $U'_q({\rm so}_4)$ of the universal
enveloping algebra $U({\rm so}_4)$ obtained by deforming the defining relations
for skew-symmetric generators of $U({\rm so}_4)$. This algebra is used in
quantum gravity and algebraic topology. We construct a homomorphism $\phi$ of
$U'_q({\rm so}_4)$ to the certain nontrivial extension of the Drinfeld--Jimbo
quantum algebra $U_q({\rm sl}_2)^{\otimes 2}$ and show that this homomorphism
is an isomorphism. By using this homomorphism we construct irreducible finite
dimensional representations of the classical type and of the nonclassical type
for the algebra $U'_q({\rm so}_4)$. It is proved that for $q$ not a root of
unity each irreducible finite dimensional representation of $U'_q({\rm so}_4)$
is equivalent to one of these representations. We prove that every finite
dimensional representation of $U'_q({\rm so}_4)$ for $q$ not a root of unity is
completely reducible. It is shown how to construct (by using the homomorphism
$\phi$) tensor products of irreducible representations of $U'_q({\rm so}_4)$.
(Note that no Hopf algebra structure is known for $U'_q({\rm so}_4)$.) These
tensor products are decomposed into irreducible constituents.
