  We establish a new Howe duality between a pair of two queer Lie superalgebras
(q(m),q(n)). This gives a representation theoretic interpretation of a
well-known combinatorial identity for Schur Q-functions. We further establish
the equivalence between this new Howe duality and the Schur-Sergeev duality
between q(n) and a central extension $\Hy_k$ of the hyperoctahedral group H_k.
We show that the zero-weight space of a q(n)-module with highest weight
$\lambda$ given by a strict partition of n is an irreducible module over the
finite group $\Hy_n$ parameterized by $\lambda$. We also discuss some
consequences of this Howe duality.
