Working down a coal mine...

Another thing that is frustrating me is trying to sort out the best/most economical way to be employed here. The main problem is that the tax and national insurance burden on employers is so heavy that schools will do anything to avoid issuing a contract of employment. There are four main ways to be employed:
on a regular contract as an employee – next to impossible to obtain because as well as paying your salary, the employer has to make substantial contributions to the social security system (INPS) on your behalf, provide sickness and holiday pay, and pay two extra (half-)month bonuses (one in the summer and one at Christmas);
on a project contract – for a fixed length of time or fixed number of hours at a fixed salary over a longer period but not exceeding 183 days a year (eg a long term temporary contract). If you do work for more than 183 days (and it's not clear whether it's actual working days or days living in Italy) then you automatically become resident for tax purposes in Italy. The employer deducts 20% of the gross as pre-payment of tax and you have to make a tax return at the end of the year (if you earn less than 30,000 euros a year) or every quarter (if you earn more) which in turn means employing an accountant. Provided you don't earn more than 5,000 euros you don't have to pay INPS but once over that amount you have to be registered. Employers like these contracts because they can write almost anything they want to into such a contract – so employee beware!;
on an occasional contract – for a fixed number of hours or one specific project only but not for regular ongoing work (ie. short-term temporary work). These contracts can only be used three times a year and for total payment not exceeding 5,000 euros. Again 20% is deducted as tax.
as an independent contractor – for this to happen you have to open a Partita IVA which is the equivalent of registering for VAT. Sounds good but here's the rub. In your first year you not only have to pay 30% of your gross income as tax but also 28% of your net income as INPS. You can deduct expenses against tax (not that there are many of these as a teacher) but effectively it means in your first year you hand over 50% of your income to the government – and, unless you are in your twenties, you will never see any benefit from your payment into the social security system (in fact most Italians believe that they will not receive a state pension worth more than peanuts even though they have paid into the system all their lives).
The other way is to use your British national insurance number and sign a declaration that you will declare all your income in the UK and pay tax on them there. This is not illegal but neither is it in the spirit of the law and again it is limited to 183 days a year.
A complication in all this is that the Italian tax year runs from January to December whereas the British tax year runs from April to March.

And if you think I'm being unduly negative, read the comments from Mr Jam. He knows what he's talking about.