Something like the following:
Map[Sort[#] &, list]
(*{{{a, 1}, {b, 3}, {c, 5}}, {{a, 5}, {b, 1}, {c, 3}}, {{a, 5}, {b, 3}, {c, 1}}, {{a, 1}, {b, 5}, {c, 3}}}*)
A first approximation:
MyOrderList[list_List, order_?(Positive[#] && Element[#, Integers] &)] :=
Block[{slist, list1, list2, list3, list4, mylist},
slist := Map[Sort, list];
list1 := Extract[Select[Tuples[slist[[1]], Length[slist[[1]]]],
ContainsAll[#, slist[[1]]] &], order];
list2 := Extract[Select[Tuples[slist[[2]], Length[slist[[2]]]],
ContainsAll[#, slist[[2]]] &], order];
list3 := Extract[Select[Tuples[slist[[3]], Length[slist[[3]]]],
ContainsAll[#, slist[[3]]] &], order];
list4 := Extract[Select[Tuples[slist[[4]], Length[slist[[4]]]],
ContainsAll[#, slist[[4]]] &], order];
mylist := List[list1, list2, list3, list4];
Return[If[order <= 6, mylist, HoldForm[MyOrderList]]];
];
Tests:
MyOrderList[list, 1]
(*{{{a, 1}, {b, 3}, {c, 5}}, {{a, 5}, {b, 1}, {c, 3}}, {{a, 5},
{b, 3}, {c, 1}}, {{a, 1}, {b, 5}, {c, 3}}}*)
MyOrderList[list, 2]
(*{{{a, 1}, {c, 5}, {b, 3}}, {{a, 5}, {c, 3}, {b, 1}}, {{a, 5},
{c, 1}, {b, 3}}, {{a, 1}, {c, 3}, {b, 5}}}*)
MyOrderList[list, 3]
(*{{{b, 3}, {a, 1}, {c, 5}}, {{b, 1}, {a, 5}, {c, 3}}, {{b, 3},
{a, 5}, {c, 1}}, {{b, 5}, {a, 1}, {c, 3}}}*)
MyOrderList[list,4]
(*{{{b, 3}, {c, 5}, {a, 1}}, {{b, 1}, {c, 3}, {a, 5}}, {{b, 3},
{c, 1}, {a, 5}}, {{b, 5}, {c, 3}, {a, 1}}}*)
MyOrderList[list,5]
(*{{{c, 5}, {a, 1}, {b, 3}}, {{c, 3}, {a, 5}, {b, 1}}, {{c, 1},
{a, 5}, {b, 3}}, {{c, 3}, {a, 1}, {b, 5}}}*)
MyOrderList[list,6]
(*{{{c, 5}, {b, 3}, {a, 1}}, {{c, 3}, {b, 1}, {a, 5}}, {{c, 1},
{b, 3}, {a, 5}}, {{c, 3}, {b, 5}, {a, 1}}}*)
Your problem in general is somewhat complicated for more characters, but I will think a little more about it.