Term for non-ARX symmetric-key cryptographic primitive design strategy
Primitives based purely on bitwise and shift/rotate operations can be described as being based on:

Binary Polynomials

Bitwise and, or, not, xor are element-wise $\mathbb{F}_2$ additions and subtractions; $n$-bit shift are based on multiplication by $x^n$ reducing $x^w$ where $w$ is the width of the word; rotation are based on multiplication reducing $x^w+1$.