In order to construct a gauge-invariant two-point function in a Yang-Mills theory, we propose the use of the all-order gauge-invariant transverse configurations Ah. Such configurations can be obtained through the minimization of the functional Amin2 along the gauge orbit within the BRST-invariant formulation of the Gribov-Zwanziger framework recently put forward in [1,2] for the class of the linear covariant gauges. This correlator turns out to provide a characterization of nonperturbative aspects of the theory in a BRST-invariant and gauge-parameter-independent way. In particular, it turns out that the poles of ?Aμh(k)Aνh(-k) are the same as those of the transverse part of the gluon propagator, which are also formally shown to be independent of the gauge parameter α entering the gauge condition through the Nielsen identities. The latter follow from the new exact BRST-invariant formulation introduced before. Moreover, the correlator ?Aμh(k)Aνh(-k) enables us to attach a BRST-invariant meaning to the possible positivity violation of the corresponding temporal Schwinger correlator, giving thus for the first time a consistent, gauge parameter independent, setup to adopt the positivity violation of ?Aμh(k)Aνh(-k) as a signature for gluon confinement. Finally, in the context of gauge theories supplemented with a fundamental Higgs field, we use ?Aμh(k)Aνh(-k) to probe the pole structure of the massive gauge boson in a gauge-invariant fashion.