The order is AX CX DX BX SP BP SI DI
My personal suspicion is that it's something to do with bit ordering. If you reverse the order of bits from LSB to MSB, then BX and CX switch places.
[1] Indexed addressing modes do need to use BX or BP as a base on 8086. Indexed modes are a bit CISCy though, considering they are often used with lea to do non-memory calculations.
Where the bits go depend on the instruction and addressing mode. They generally follow this pattern; some addressing modes don't support some registers though.