Weak, rotated magnetic and radiofrequency quadrupole fields in electron guns and injectors can couple the beams horizontal with vertical motion, introduce correlations between otherwise orthogonal transverse momenta, and reduce the beam brightness. This paper discusses two important sources of coupled-transverse dynamics common to most electron injectors. The first is quadrupole focusing followed by beam rotation in a solenoid, and the second coupling comes from a skewed high-power RF coupler or cavity port which has a rotated RF quadrupole field. It is shown that a DC quadrupole field can correct for both types of couplings and exactly cancel their emittance growths. The degree of cancellation of the RF skew quadrupole emittance is limited by the electron bunch length. Analytic expre ssions are derived, and compared with emittance simulations and measurements. Show Partial Abstract