I think you may be thinking of particle spin as something spinning, so it would have angular momentum, so from there you assume there is energy and its somehow conserved. But I don't think anything is actually spinning.
Also, think about what would be happening if they were actually spinning, like a wheel. If you have a spinning wheel and magically split it in half perpendicular to the axis of rotation, you have the same energy in the system, the two parts still spin in the same direction, one doesn't suddenly start spinning in the other direction.
Or take the other way of looking at that. You have a pair of wheels. both spinning in opposite directions, and join the axles together.The energy doesn't cancel out, it sums and gets converted into a force twisting the axle and eventually into heat.