Changing scan order of Gibbs Sampler on each iteration I'm implementing an algorithm that requires the use of Gibbs Sampling and, due to the nature of the way I store the values, it would be efficient to change the order of the updates on each component at every iteration.
For example, if I have three components, the first iteration will be over [1, 2, 3], but the next one could be [2, 3, 1], then [1, 3, 2], and so on.
If the answer is "it depends" then I'll edit the question with more details.

One can consider a Gibbs sampler as a combination of componentwise Markov kernels $G_i$ with $$G_i(\theta,\theta^\prime)=p(\theta^\prime_i|\theta_{-i})\delta_{\theta_{-i}}(\theta^\prime_{-i})$$Since each kernel is stationary wrt the joint distribution, so is the combination. The only major constraint is ensuring that the combination results in an irreducible Markov chain, meaning that all components must move infinitely often. Other than that, considering an arbitrary sequence $(i_n)_n$ over $\{1,2,\ldots,p\}$ leads to a valid Gibbs sampler
$$G_{i_m}\circ\cdots\circ G_{i_2}\circ G_{i_1}$$
