How to divide a list into several single loops

ClearAll[consecutiveRuns]

rule = a : {__} /; MatchQ[{1 ..} | {-1 ..}] @ Differences[a] :> a;

consecutiveRuns = SequenceSplit[#, rule] &;

Examples:

consecutiveRuns @ {4, 3, 2, 1, 7, 6, 5}

{{4, 3, 2, 1}, {7, 6, 5}}

consecutiveRuns @ {3, 2, 1, 7, 6, 4, 5}

{{3, 2, 1}, {7, 6}, {4, 5}}

For the example in flinty's answer:

consecutiveRuns @ {1, 2, 3, 4, 5, 2, 3, 0, 2, 1, 3, 1, 2, 3, 4, 3, 2, 1}

{{1, 2, 3, 4, 5}, {2, 3}, {0}, {2, 1}, {3}, {1, 2, 3, 4}, {3, 2, 1}}

updownruns[list_] := 
 Module[{s1 = Split[list, Abs[#1 - #2] == 1 &]}, 
  Reap[Do[If[SameQ @@ Differences[s], Sow[s], 
      Sow /@ TakeDrop[s, 1 + Length[First@Split@Differences@s]]
      ], {s, s1}]][[2, 1]]]

This handles cases like this:

updownruns[{3, 2, 1, 7, 6, 4, 5}]
(* result: {{3, 2, 1}, {7, 6}, {4, 5}} *)

But also cases like this:

updownruns[{1, 2, 3, 4, 5, 2, 3, 0, 2, 1, 3, 1, 2, 3, 4, 3, 2, 1}]
(* result {{1, 2, 3, 4, 5}, {2, 3}, {0}, {2, 1}, {3}, {1, 2, 3, 4}, {3, 2, 1}} *)