Hi David. I see no reason that you would obtain a new assembly with the same orientation and starting point as the previous one. It depends on the approach followed by the assembler. Note that, technically speaking, a circular genome has no starting point. Although it is not required, for better comparisons, you can manually rotate your new assembly to make its starting point match the previous assembly. You can use show-coords from MUMmer to check how many bases you should rotate the assembly (should be equal to the size of your small aligned fragment).
I would also suggest to use dnadiffwhich is also part of MUMmer package. It reports SNPs and short INDELs, which can help identify points of variability between your assemblies.
EDIT: If you know the exact number of bases to rotate the circular genome, you can use seqkit restart as described here to rotate the fasta sequence.
If these are complete bacterial genomes, they are likely circular. I believe you are seeing this dot plot because one the genomes has the opposite orientation (reverse complement) and is rotated compared to the reference. I don't think there is any rearrangement.