Cyclically sort lists of mixed element types?

Here is a function

cyc[list_] := RotateLeft[list, First@Ordering[list, 1]]

For your lists:

list1 = {1, 2, a[1], 8, b[4], 9};
list2 = {8, b[4], 9, 1, 2, a[1]};
cyc[list1] == cyc[list2]
True