What's the word for converting a positive number to negative one and negative to positive?

Negation absolutely is is the correct term to use here: it is a synonym of "additive inverse", as noted in the second sentence of its Wikipedia article. This is a well-defined mathematical operation which would absolutely be understood in a numeric context, and is used constantly in math classes from grade school through Master's programs; the dictionary just wasn't providing the domain-specific definition of the term.

The action would be called "Finding the additive inverse" of the number.

The additive inverse is the number that, when added to your initial number, adds up to zero.

There are other possible terms in the Wikipedia excerpt below

In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the opposite to a positive number is negative, and the opposite to a negative number is positive. Zero is the additive inverse of itself.

Of note is "opposite number".

The opposite of a number is just the number on the opposite side of zero on the number line.