When the top input is high, T1 acts as an emitter follower and its emitter is at approximately 5V-Vbe. T2 is not conducting. When it is low, T2 acts as an emitter follower and its emitter is at approximately GND+Vbe, and T1 is not conducting.
Similarly for the other input, with T3 and T4 respectively.
Note that as emitter followers T2 and T4 don't turn on unless their bases are actively pulled towards GND.
Consider the following 4 cases:
A) HIGH/LOW - T1 emitter pulls to 5-Vbe. T4 emitter pulls to GND+Vbe. Coil receives +5V-(2*Vbe).
B) LOW/HIGH - T2 emitter pulls to GND+Vbe. T3 emitter pulls to 5V-Vbe. Coil receives -(+5V-(2*Vbe)).
C) LOW/LOW - T2 emitter pulls to GND+Vbe, T4 pulls to GND+Vbe. Both ends of coil at the same potential.
D) HIGH/HIGH - T1 emitter pulls to 5V-Vbe, T3 emitter pulls to 5V-Vbe. Both ends of coil at the same potential.