Electron motion and harmonic generation are investigated in the crossed-beam laser-accelerator scheme in a vacuum. Exact solutions of the equations of motion of the electron in plane-wave fields are given, subject to a restricted set of initial conditions. The trajectory solutions corresponding to axial injection are used to calculate precise emission spectra. Guided by hindsight from the analytic investigations, numerical calculations are then performed employing a Gaussian-beam representation of the fields in which terms of order epsilon(5), where epsilon is the diffraction angle, are retained. Present-day laser powers and initial conditions on the electron motion that simulate realistic laboratory conditions are used in the calculations. The analytic plane-wave work shows, and the numerical investigations confirm, that an optimal crossing angle exists, i.e., one that renders the electron energy gain a maximum for a particular set of parameters. Furthermore, the restriction to small crossing angles is not made anywhere. It is also shown that energy gains of a few GeV and energy gradients of several TeV/m may be obtained using petawatt power laser beams.