Is there a kind of relay that consumes power only when switching?
Yes, they go by various names such as a bistable relay, latching relay, or impulse relay. Common in small (telecom style 2A or less) and also some power relays have this function. 

There are various methods of actuating them, the small ones typically have either two coils (pulse one coil for 'on', pulse the other for 'off') or single coil (reverse polarity pulses for on vs. off). 

Some of the power ones are alternating (pulse on, pulse off). One application of the high-power relays is for electrical metering where the power can be remotely switched off in case of bill non-payment by the customer.