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We investigate a class of brickwork-like quantum circuits on chains of $d-$level systems (qudits) that share the so-called `dual unitarity' property. Namely, these systems generate unitary dynamics not only when propagating in the time direction, but also when propagating in the space direction. We consider space-time homogeneous (Floquet) circuits and perturb them with a quenched single-site disorder, i.e. by applying independent single site random unitaries drawn from arbitrary non-singular distribution over ${\rm SU}(d)$, e.g. one concentrated around the identity, after each layer of the circuit. We identify the spectral form factor at time $t$ in the limit of long chains as the dimension of the commutant of a finite set of operators on a qudit ring of $t$ sites. For general dual unitary circuits of qubits $(d=2)$ and a family of their extensions to higher $d>2$, we provide explicit construction of the commutant and prove that spectral form factor exactly matches the prediction of circular unitary ensemble for all $t$, if only the local 2-qubit gates are different from a SWAP (non-interacting gate). We discuss and partly prove possible extensions of our results to a weaker (more singular) forms of disorder averaging, as well as to quantum circuits with time-reversal symmetry, and to computing higher moments of the spectral form factor.