  Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.
