You don't have to go in on your day off, unless that is specified within your contract.
In fact, if you work in an LEA school, then there is a specific statement in the STCPD which states that a part-timer cannot be asked to work on a day that they don't normally work.
In an academy school, or independent school, then they would have to specify it in the contract eg. some independent schools specifically state that a teacher will be required to work Saturday mornings (not a normal working day) or do evening duty (not a normal time for working).
It never ceases to amaze me how this idea that you have to do "0.6 of INSET if you are 0.6" gets trotted out! It's not really the fault of teachers, because a lot of Heads seem to believe this, and they pass this myth on.
If all the INSET days were on a day you don't work, then you'd end up doing 3 extra unpaid days! It also goes the other way, if INSET days all land on a day you normally work, then you should go in for all of them, otherwise you're getting paid for 2 days that you haven't worked.
I stress though, that if they specifically state in the contract "you must do 0.6 of all INSET days, even if they land on your day off", and you signed it, then at the point of signing, you have agreed to it.
Another word of caution: it may be easier and safer for your career to find out your Head's view and just go along with it, and look for work elsewhere if you don't like it. Some heads can be very vindictive if you try to question their authority, and you will need a reference from them when you move to your next school.