  We establish necessary and sufficient conditions for invertibility of
symmetric three-by-three block matrices having a double saddle-point structure
\fb{that guarantee the unique solvability of double saddle-point systems}. We
consider various scenarios, including the case where all diagonal blocks are
allowed to be rank deficient. Under certain conditions related to the nullity
of the blocks and intersections of their kernels, an explicit formula for the
inverse is derived.
