Ubuntu Networking :: Checking For Open Ports Remotely

I'm trying to set up an SSH connection from my school to my home, but not on port 22. I originally tried port 2222, but it didn't work. I called my school IT people and they said they block that port. I asked if they care if I set up an SSH and they said no, but that they wouldn't tell me which ports are blocked and which are open for "security" reasons (which I guess I can actually understand). They suggested just using port 22 or 222, but said if a ports open I can use it.

My question is, can I check ports without setting up SSH? It seems like a hassle to try a different port every day on my home SSHD file, come to school, see if it works and repeat. Is there a way I can check my computer home for connections that could connect, even if there isn't a service listening? I don't know how to do that, or even if its possible. If not, I suspect I'll just try a few until I find something that works, or just go ahead and use 22.