  A full intrinsic quadric is a normal complete variety with a finitely
generated Cox ring defined by a single quadratic relation of full rank. We
describe all surfaces of this type explicitly via local Gorenstein indices. As
applications, we present upper and lower bounds in terms of the Gorenstein
index for the degree, the log canonicity and the Picard index. Moreover, we
determine all full intrinsic quadric surfaces admitting a K\"ahler-Einstein
metric.
