Washing machine pressure switches directly control the hot and cold water inlet solenoids and so are rated for quite high currents - 15A at 230V AC is typical. So for low-voltage DC applications, the switches can certainly cope with (say) 5A. This means that, for most loads, a relay won't be needed.
During ‘on’ time of spin direction timer IC2, the output of negative- edge triggerd JK flip-flop at pin 2 goes low to energise relay RL2 and washing machine motor rotates in one direction. During the off time of IC2, the output of N1 goes high again to de-energise relay RL1, which cuts off the mains supply to RL2 and the monitor stop rotating. Floating point trouble may occur at trigger pin 2 of IC1. Resister R8 over comes this problem by holding pin 2 high.