Geometric properties of operators of quantum Dirac constraints and physical observables are studied in semiclassical theory of generic constrained systems. The invariance transformations of the classical theory -- contact canonical transformations and arbitrary changes of constraint basis -- are promoted to the quantum domain as unitary equivalence transformations. Geometry of the quantum reduction of the Dirac formalism to the physical sector of the theory is presented in the coordinate gauges and extended to unitary momentum-dependent gauges of a general type. The operators of physical observables are constructed satisfying one-loop quantum gauge invariance and Hermiticity with respect to a physical inner product. Abelianization procedure on Lagrangian constraint surfaces of phase space is discussed in the framework of the semiclassical expansion.