It is built up the Hamiltonian quantization of Yang-Mills theories in the general homogeneous axial gauges of the typenμAμ=0 (n 0≠0). A (tentative) physical Hilbert space is defined, where the Gauss law holds in a weak way, with the Poincaré covariance as well. The role of the residual gauge symmetry with respect to the unphysical degrees of freedom is clarified. The necessity of zero inner-square “physical” state-vectors is explained as owing to an interplay between the recovering of Poincaré covariance and the related lasting of the residual gauge symmetry. A unified well-defined prescription for the (photon) gluon propagator is deduced, leading to an extension of the one originally proposed by Leibbrandt and Mandelstam for the specific lightlike gauge.