Hi Killmon and benitacanova --
Just to clarify, taxes are not optional here in Spain. Either you are working legally or you are not. Legally working means that either you are paying your taxes as an autonomo or your employer is paying them on your behalf.
Regarding social security payments. If you are autonomo the company that has contracted you cannot be held responsible for paying your social security as long as the agreement you signed with them was in good faith. However, if you should get injured on the premise of a business while on the job and you don't have social security coverage, the company where you go to teach could be held responsible for paying your medical expenses.
While your hiring company cannot withhold your payment legally, unless it was stated otherwise in your agreement with them, they can make it a stipulation of working with them and terminate your agreement at will.
You ought to be glad to be working for this company as it is one that apparently takes working legally seriously. Remember that if you are in fact a bonafide autonomo, no one is responsible for your vacation time except for you... and there is no maternity leave for autonomos unless you pay a higher amount to social security. You are autonomo, self employed... no one is responsible for your expenses except for you.
BTW it could be that they company is requiring proof a payment from you because that is what their clients require. It's not the first time that I have heard of this.