I'll add the best way to remember what occurs in each device is our good old pal: ELI the ICE man.
An inductor's energy is proportional to the square of its current. No work needs to be done to change its voltage. But, work must be done to change an inductor's current. In an inductor, voltage can change abruptly, but current can only change gradually. In a physical inductor, some capacitance is present across the turns in the winding, so that a little work is needed to change the voltage.
The capacitor is the counterpart of the inductor. Its stored energy is proportional to the square of the voltage. In a cap, current may change abruptly, whereas voltage must change gradually. Work needs to be done to change its voltage, but not its current. In a real cap, there is always some inductance, so that some work is done changing the current.
Off the top of my head, that is a brief overview. More detail can be added, but that is the basic reason.