Doppler-free two-photon absorption spectra of the A(1)A(u)<--X(1)A(g)(pi*-n) transition of trans-glyoxal have been measured by means of two-photon absorption spectroscopy with counter-propagating light beams of identical photons within an external cavity. The relative energies of transition lines are measured with accuracy better than 0.0001 cm(-1). Rotational Lines are fully resolved, and 1809 Lines of the A(1)A(u)(v(7) = 1)<--X(1)A(g)(v = 0) transition are assigned for J= 0 - 75,K= 0 - 17. Rotational constants of the A(1)A(u)(v(7)=1) and X(1)A(g)(v = 0) states are determined by a least-squares fitting of eigenvalues of the A-reduced rotational Hamiltonian to energies of the assigned lines. Energy shifts, intensity anomalies, and Line splittings are observed for several lines. When an external magnetic field is applied, remarkable changes are observed for these Lines. They are identified as originating from perturbations between the A (1)A(u) and a(3)A(u) states, which become appreciable when perturbing levels are close in energy. Splittings into three lines are observed for strongly perturbed levels of I = 1, and these splittings are identified as the hyperfine splitting caused by mixing of the a(3)A(u) state.