I would have thought that as long as you are drinking plenty of water throughout the day, as well as your spinning class, then the exercise would be good for you as it will help to sweat out the toxins and to get your liver back up and running?
I remember reading ages ago that the best thing for a hangover was a swift walk / jog first thing in the morning as it kickstarts your liver and starts cleansing your body quicker.
Certainly, I used to head for the gym first thing on a Sunday morning after a skinful the night before (I am 7months pregnant at the moment so neither drinking nor going to the gym at the mo!) and would feel fine for the rest of the day. Just ensure you are drinking enough water, as your body will already be dehydrated from the alcohol.