We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely $\textsf{Max-DICUT}$, for which random ordering makes a provable difference. Whereas a $4/9 \approx 0.445$ approximation of $\textsf{DICUT}$ requires $Ω(\sqrt{n})$ space with adversarial ordering, we show that with random ordering of constraints there exists a $0.48$-approximation algorithm that only needs $O(\log n)$ space. We also give new algorithms for $\textsf{Max-DICUT}$ in variants of the adversarial ordering setting. Specifically, we give a two-pass $O(\log n)$ space $0.48$-approximation algorithm for general graphs and a single-pass $\tilde{O}(\sqrt{n})$ space $0.48$-approximation algorithm for bounded degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require $Ω(\sqrt{n})$-space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erdös-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is $\textsf{Max-CUT}$ where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for $o(\sqrt{n})$ space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints.