If you first plugged it in and then shut it down, the screen will not show anything as the phone is shut off. It IS still charging though. If you were concerned about unplugging it when fully charged, DON'T worry about it. Once it is charged, charging will stop (the phone itself controls its charging process, not the power adaptor) and it will NOT over charge (no SMART Lithium power cell will - that is part of the international SMART Lithium battery specifications - built in overcharge limiting and control).
Unplugging it when off should not trigger a power up though, that is odd.
If shut down when unplugged (i.e. on battery), and then plugged in, it should start up as that would be normal.
BTW, there is no need at all to shut it off to charge it as that makes no difference with the battery at all. Most people basically never shut off their iPhones or iPads - they just charge them as needed or as convenient to do so and let them sleep when plugged in (or use them while charging, which is fine too).