Rocket works on the principle of conservation of momentum. The rockets gain momentum equal to the momentum of the gas expelled from the engine, but in the opposite direction.
The working principle of a washing machine is centrifugation.
Centrifugation is a separation process which uses the action of centrifugal force to promote accelerated settling of particles in a solid-liquid mixture.
The washing machine consists of centrifuge for this purpose. A centrifuge is a piece of equipment that puts an object in rotation around a fixed axis, applying a force perpendicular to the axis of spin that can be very strong.