No, that would cost more energy than we extracted from it. However, what we call “spent fuel” isn’t fully used up. It can no longer be used in a normal reactor, but reactors can be build which can continue extracting energy from it. In fact, only about 2% of the usable energy has been extracted from the fuel by the time we start calling it waste. Most of the high–level waste from today’s reactors is really just fuel for reactors that we haven’t built yet, due to over–regulation.

In principle, this might be possible since any reaction can in principle be run in reverse, given a sufficient supply of energy.

In practice, I don't think this would be at all viable, and certainly would not be competitive with other much more straightforward forms of energy storage.

It's an operation cycle of very specific reactors, if I remember right its a variation of breeder reactors.