Simultaneous longitudinal and transverse focusing can be obtained in a drift tube linac by alternating the sign of the synchronous phase. It is found both theoretically and numerically that, in the alternating-phase-focused (APF) linac with a symmetric synchronous phase sequence, the lowest-order resonance due to the synchrobetatron coupling naturally occurs, causing significant emittance transfer between the longitudinal and transverse motions. Equations for the averaged APF motion are derived to investigate the coupling effect. Two approximate invariants are obtained. Results from computer simulations based on a modified parmila code yield good agreement with the formulas for the invariant and tune shift derived from the theory. We then suggest a way to move the parameters away from the lowest-order resonance. Simulation results for the effect of space charge on the emittance exchange along with the synchrobetatron coupling are briefly discussed.