  Let $d\in\mathbb{N}$ and let $\varphi\colon(0,1)\to[0,d]$. We prove that
there exists a set $F\subset\mathbb{R}^d$ such that
$\operatorname{dim}_A^\theta F=\varphi(\theta)$ for all $\theta\in(0,1)$ if and
only if for every $0<\lambda<\theta<1$, \[0\leq
(1-\lambda)\varphi(\lambda)-(1-\theta)\varphi(\theta)\leq
(\theta-\lambda)\varphi\Bigl(\frac{\lambda}{\theta}\Bigr).\] In particular, the
following behaviours which have not previously been witnessed in any examples
are possible: the Assouad spectrum can be non-monotonic on every open set, and
can fail to be H\"older in a neighbourhood of 1.
