Taking VAT out of a price is not taking the rate off it
This is the mistake the page exists to stop. If a price already includes value added tax at a fifth, the tax inside it is not a fifth of that price. The tax was worked out on the amount before tax, so the price you are looking at is six fifths of what the seller actually charged. Take a fifth off the gross and you are taking a fifth of a bigger number: you end up below the real net figure, and the shortfall is a fifth of a fifth.
Working backwards means dividing rather than subtracting. The price before tax is the price including tax divided by one plus the rate. On a round hundred at a fifth, adding the tax gives a hundred and twenty and the tax is twenty; taking the tax back out of a hundred and twenty returns exactly a hundred, while subtracting a fifth would leave ninety-six. The direction selector at the top of the form is which of those two questions you are asking, and the page answers whichever one you pick with all three figures showing, because seeing the before, the tax and the after together is what makes the answer checkable.
Where the rates come from
Every rate offered here is one a Member State has reported to the European Commission, read from the Commission's own database through the web service it publishes for the purpose. Nothing on this page is a rate somebody remembered or copied from another calculator. Picking a country fills its standard rate into the field and names the source and the day that filing took effect underneath; the field stays editable, so you can type a rate the file does not carry and the page will still do the arithmetic.
The buttons beside the field are the other rates that country applies. A Member State may run reduced rates below its standard one, and some run a lower super-reduced rate or keep a parking rate for goods that would otherwise be standard-rated. Where a country files the same percentage under two of those labels, it appears once: what you need to know is that the percentage is available, not which drawer the Commission files it in. One Member State applies no reduced rate at all, and the page simply offers nothing beside its standard rate rather than inventing one.
What a rate table cannot tell you
Knowing that a country applies a reduced rate is not the same as knowing your goods qualify for it. That question is decided by category lists in the VAT Directive, which run to two dozen headings covering food, water, medicines, books, passenger transport, housing and much else, and each Member State chooses which of them it applies a reduced rate to and at what level. That mapping is enormous — for a single country the full list of customs codes runs to tens of kilobytes — and it is not in this page. What is here is the levels; the Commission's database is where to check the category.
Territorial rates are left out on purpose, and the page tells you when it has left some out and how many. Several Member States apply special rates in particular regions: an island, an overseas department, a territory with its own historical arrangement. Those rates are perfectly real and they are not the rates that apply in the country at large, so listing them beside the national ones would make a country look as though it had far more rate levels than it does. One Member State also files a rate for a territory that levies a tax of its own rather than VAT at all; that figure is named and set aside rather than offered, because charging it as VAT would be charging the wrong tax.