  In this paper we describe the geometry of the 2m-dimensional Fano manifold G
parametrizing (m-1)-planes in a smooth complete intersection Z of two quadric
hypersurfaces in the complex projective space P^{2m+2}, for m>0. We show that
there are exactly 2^{2m+2} distinct isomorphisms in codimension one between G
and the blow-up of P^{2m} at 2m+3 general points, parametrized by the 2^{2m+2}
distinct m-planes contained in Z, and describe these rational maps explicitly.
We also describe the cones of nef, movable and effective divisors of G, as well
as their dual cones of curves. Finally, we determine the automorphism group of
G. These results generalize to arbitrary even dimension the classical
description of quartic del Pezzo surfaces (m=1).
