Hybridisation and the Schrödinger equation
Mathematically, atomic/molecular orbitals are 1-electron wavefunctions (hydrogen-like wavefunctions) that are used as a basis with-which the total N-election wavefunction is expanded.  The N-election wavefunction is a determinant (or a linear combination of determinants) built from these 1-electron wavefunctions: an anti-symmetric (Fermi statistics) linear combination of Hartree products (usually a product of MOs expanded in a basis of AOs).

For simplicity's sake, lets assume we are interested only in 1-determinant wavefunctions.  The Hartree-Fock method will (when used where HF is appropriate) result in a variationally-optimal 1-determinant wavefunction that is an energy eigenfunction of the applicable Hamiltonian operator.  Not only is the final, self-consistently optimized HF total N-electron wavefunction an energy eigenfunction, but the individual HF canoncial MOs are also eigenfunctions of the Hamiltonian/Fock operator, with eigenvalues that are the so-called 'orbital energies' (canonical orbitals are defined as the optimzed orbitals which diagonalized the Fock matrix).

However, there is nothing special about these canonical orbitals in the context of the total system energy.  Any linear, norm-preserving transformation of the occupied orbital space will result in an N-electron wavefunction that is still an eigenfunction of the Fock operator, while the individual MOs are no longer eigenfunctions of the Fock operator (for example, localized MOs, LMOs).

So, once we have a HF wavefunction and optimized MOs, we can linearly transform the MOs however we want.  Such a linear transformation might have the effect of producing orbitals that resemble hybridized orbitals.  Or maybe we are interested in generating localized MOs.  The point is, a linear transformation within the occupied orbital space results in a wavefunction that has the exact same electon density and energy eigenvalue as the wavefunction HF originally provides us in terms of Canonical orbitals.

Orbitals transformed to look like hybrid-orbitals are on the exact same footing as any other choice of orbital.