For every action there is an equal and oposite reaction (Newton's Law). Although it is originally for inertia, it also applies here. The rules are set in place for the very few that abuse the system. I'm sure even HR agrees that the 24 hour rule is rediculous. I'm pretty sure if you speak with them, your situation is a tad bit different and they would be willing to work with you.
IF that is not the case, here is my recommendation: If you start work at 6:00am, I would call in "every day" at 5:50am and say "I MIGHT, not be in tomorrow". Do this for about a week and I'll bet they change their ways. Your simply following THEIR rules on this one. I know it is nit picky, but sometimes it takes one person to point out the obvious before action is taken to improve it.
I have OCD "Obsessive Crappie Disorder"