The order in which you update the new solution has a big impact on the values produced from the initial data to the first approximation. Some updated values will be the same as the jacobi iteration, while others will contain the latest information a la Gauss-Seidel. Given a specific ordering, there may be two values that are the same if the initial vector $x_0$ has non-unique values. But for two different orderings, if the the solution vector is large, and the order of updating the gauss-seidel formula is random, the more unlikely it is that you will produce similar 1st iterations. I'm not sure if there is a proof for that different orderings produce unique 1st iterations, but even if it is possible, the probability is very unlikely.

The order in which you update the new solution has a big impact on the values produced from the initial data to the first approximation. Some updated values will be the same as the jacobi iteration, while others will contain the latest information a la Gauss-Seidel. Given a specific ordering, there may be two values that are the same if the initial vector $x_0$ has non-unique values. But for two different orderings, if the the solution vector is large, and the order of updating the gauss-seidel formula is random, the more unlikely it is that you will produce similar 1st iterations. I'm not sure if there is a proof for that different orderings produce unique 1st iterations, but even if it is possible for them to be non-unique, the probability is very unlikely.