Is Gibbs Sampling affected by the order in which the conditional probabilities are calculated? As given, first x_1,(t+1) is calculated from P( x_1,t | x_j,t ) where j = 2:K; then x_2,t conditioned on x_1,(t+1) and x_j,t for j = 3:K and so on.

Would things change much (convergence rate, accuracy etc) if the order was changed, say starting with x_K,(t+1)?

Does detailed balance have any bearing on this question?