Is this something like what you need?
sequence = {6, 3, 2, 4, 1, 2, 3, 4, 5, 4, 3, 2, 1, 3}; Do[
s = sequence[[a ;; b]];
If[s == Reverse[s], Print[s]], {a, 1, Length[sequence] - 1}, {b,
a + 1, Length[sequence]}]
{1,2,3,4,5,4,3,2,1}
{2,3,4,5,4,3,2}
{3,4,5,4,3}
{4,5,4}
