  We address a class of schemes for the Euler equations with the following
features: the space discretization is staggered, possible upwinding is
performed with respect to the material velocity only and the internal energy
balance is solved, with a correction term designed on consistency arguments.
These schemes have been shown in previous works to preserve the convex of
admissible states and have been extensively tested numerically. The aim of the
present paper is twofold: we derive a local total energy equation satisfied by
the solutions, so that the schemes are in fact conservative, and we prove that
they are consistent in the Lax-Wendroff sense.
