The UK flag is NOT entirely symmetrical!
You can rotate the flag and it still will be the right way up but you can't invert it.
Look at the white diagonals and you will see why.
On the side next to the flagpole, (the hoist side), the thick white band is above the red band on both diagonals, the white band being part of the cross of St Andrew, the Scottish flag and the red band being part of the cross of St Patrick, the Irish flag.
The cross of St Andrew is above that of St Patrick at the hoist because the cross was added to the flag before St Patrick's cross.
On the side that flutters free, the fly side, this is reversed, with the Irish flag being above the Scottish flag.