We show that if E is an equivalence of upper semicontinu-
ous Fell bundles B and C over groupoids, then there is a linking bundle
L(E ) over the linking groupoid L such that the full cross-sectional alge-
bra of L(E ) contains those of B and C as complementary full corners,
and likewise for reduced cross-sectional algebras. We show how our re-
sults generalise to groupoid crossed-products the fact, proved by Quigg
and Spielberg, that Raeburn's symmetric imprimitivity theorem passes
through the quotient map to reduced crossed products.