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
TortoiseGit is optimized for Git for Windows.
However, starting with TortoiseGit 1.8.13.0 Cygwin git can also be used with TortoiseGit.
You have to configure it as follows on TortoiseGit settings: Open TortoiseGit settings dialog and open the General page.
1) Select the [CYGWIN-INSTALL-PATH]\bin-folder as git.exe folder.
2) Configure the HOME environment variable in Windows, so that Cygwin and TortoiseGit are using the same home directory and global git-config. Use the normal Windows notation here (e.g., "C:\Users\USERNAME"). By default, TortoiseGit uses the Windows home directory which is normally located under c:\Users and Cygwin uses its own home directories which are located under [CYGWIN-INSTALL-PATH]\home.
3) Configure AutoCrLf, this is necessary as TortoiseGit and Cygwin Git have different defaults. The default in Cygwin Git is true.
4) Go to TortoiseGit the section called “Advanced Settings” and set CygwinHack to true in order to activate cygwin workarounds.
5) Reboot.
Based on https://tortoisegit.org/docs/tortoisegit/tgit-dug-settings.html