[...] so long as you don't have to run the computation through multiple rounds (like having 32bit ALU but doing 64bit arithmetic) That does not sound right, yo

[...] so long as you don't have to run the computation through multiple rounds (like having 32bit ALU but doing 64bit arithmetic) That does not sound right, your byte ordering should not affect the ALU, it will always perform the same operations. If you are doing a multi-word add, you have to add from least significant to most significant word because of the carry. And the ALU has no idea what you are adding, whether the numbers are independent or part of a multi-word integer. At best I could imagine that there might be some impact when fetching operands as in big-endian you have to fetch from decreasing addresses which might be less efficient then fetching from increasing addresses.