I have recently installed Cygwin along with the Git binaries and a copy of TortoiseGit on a fresh Windows XP install. I seem to be able to do most Git operations through Tor
I tried those solutions, but they didn't work for me. What did work was to move all GIT related executables in the cygwin directory into a sub folder not in my path, then I installed the latest version of GIT from http://git-scm.com/, added the location of that installed GIT executable to my path, killed all instances of Windows Explorer & bash windows, launched a new instance of Windows Explorer, and then, I was able to commit via TortoiseGit. Currently (March 4, 2014) the cygwin version of GIT is 1.7.9, whereas the latest version from http://git-scm.com/ is 1.9.0. That seems to make a difference.