  We introduce a two-parameter function $\phi_{q_+,q_-}$ on the infinite
hyperoctahedral group, which is a bivariate refinement of the reflection length
keeping track of the long and the short reflections separately. We provide a
complete characterization of the parameters $q_+,q_-$ when the signed
reflection function $\phi_{q_+,q_-}$ is positive definite and we prove that
this condition holds if and only if $\phi_{q_+,q_-}$ is an extreme character of
the infinite hyperoctahedral group. We construct the corresponding
representations as a natural action of the hyperoctahedral group $B(n)$ on the
tensor product of $n$ copies of a vector space, which gives a two-parameter
analog of the classical construction of Schur--Weyl.
  We apply our characterization to construct a cyclic Fock space of type B
which generalizes the one-parameter construction in type A found previously by
Bo\.zejko and Guta. We also construct a new cyclic Gaussian operator of type B
and we relate its moments with the Askey--Wilson--Kerov distribution by using
the notion of cycles on pair-partitions, which we introduce here.
