  We study a gauge-invariant renormalization scheme (GIRS) for composite
operators, regularized on the lattice, by extending the coordinate space
(X-space) scheme proposed some years ago. In this scheme, Green's functions of
products of gauge-invariant operators located at different spacetime points are
considered. Due to the gauge-invariant nature of GIRS, gauge fixing is not
needed in the lattice simulations. Also, when operator mixing occurs, the
gauge-variant operators (BRST variations and operators which vanish by the
equations of motion) can be safely excluded from the renormalization process.
  We propose a number of variants of GIRS, including integration over time
slices of the operator insertion point in a Green's function, which may lead to
reduced statistical noise in lattice simulations. We employ these variants in
the renormalization of fermion bilinear operators and the study of mixing
between the gluon and quark energy-momentum tensor operators. We extract the
one-loop conversion factors relating the nonperturbative renormalization
factors in different versions of GIRS to the reference scheme of $\overline{\rm
MS}$.
