Here's what I did to do some testing:
1. Created a new double-sided doc with automatic text frames on 4 pages.
2. Selected text frame on the first page > Sample text, to add some text.
3. F3 to create a new paragraph style which I named TabTest for which I set first line indent to 6mm, and added a left tab to 12mm.
4. Selected text frame on the first page again > F2 (Properties palette) > Text > Style Settings > TabTest. The first line of every paragraph gets indented at 6mm, and the left tab I created is exactly at 12mm as it was set in TabTest paragraph style. If you change 12mm to 6mm for the left tab (so that it matches first line indent), it stays at 6mm throughout the document, no moving around.
Please try doing these steps and tell us where the problem occured.