Don't cut it back if the weather is still cold, wait till whenever your spring starts. If its like here, where spring is a bit hit and miss, being warm one minute and freezing the next, don't cut until it's settled into a warmer spell -by then, if the plant has survived, you should be seeing new growth anyway, and you then cut back to where the new growth is.

I think Bamboo must be an optimist. (and why not?) But if by 'harsh' you mean quite sharp frosts, it isn't terribly likely that your Bougainvillea will have survived. These are Mediterranean plants and like cool wet winters and hot warm summers. By all means cut it back, but if it's been frosted don't expect much! My experience of growing them has been in conservatories, but we've always lost them even though they've been protected from frost. They didn't like the cold temperatures in winter.

I will admit I was avoiding say it's probably an ex bougainvillea, Bertiefox - you never know your luck, fingers crossed...