  We ask the question whether for a given unitary representation $U$ the
associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{c\, c})$ has spectrum
invariant under inversion -- in this case we say that $\rho_{U}$ has symmetric
eigenvalues. This does not have to be the case. However, we give affirmative
answer whenever a certain condition on the growth of dimensions of irreducible
subrepresentations of tensor powers of $U$ is imposed. This condition is
satisfied whenever $\widehat{\mathbb{G}}$ is of subexponential growth.
