Is there a canonical form that uses AND and XOR?
You're looking for Reed-Müller forms. The introduction to this paper summarizes some families of these forms:

An arbitrary n-variable  function $f(x_1,x_2,...,x_n)$ can be represented using the positive polarity Reed-Muller form (PPRM)
$f(x_1,x_2,...,x_n) = a_0\oplus a_1x_1\oplus a_2x_2\oplus\ldots\oplus a_{12}x_1x_2\oplus a_{13}x_1x_3\oplus\ldots\oplus a_{n-1,n}x_{n-1}x_n\oplus\ldots\oplus a_{12\ldots n}x_1x_2\ldots x_n$.
For each function $f$, the coefficients $a_i$ are determined uniquely, so PPRM is a canonical form. If we used either only the positive literal ($x_i$) or only the negative literal ($\overline{x_i}$) for each variable in Eq. (4)1, we get the FPRM. There are $2^n$ possible combinations of polarities and as many FPRMs for any given logical function.
If we freely choose the polarity of each literal in Eq. (4), we get a GRM. In GRMs, contrary to FPRMs, the same variable can appear in both positive and negative polarities. There are $n2^{n-1}$ literals in Eq. (4), so there are $2^{n2^{n-1}}$ polarities for an $n$-variable function and as many GRMs. Each of the polarities determines a unique set of coefficients, and thus each GRM is a canonical representation of a function.


1 It's not clear from the text, but equation 4 is the generic PPRM equation included in the quote