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Is there a way to work with angles in DMS without converting them to decimal degrees, that is to sum them, subtract them?

I do not know how to even enter it without using string. My idea is something like: 20°32'15.828'' - 22°32'12.8''.

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You may use either MixedRadixQuantity or Quantity with MixedMagnitude and MixedUnit.

In both cases the units you need are {"AngularDegrees", "ArcMinutes", "ArcSeconds"}. 

p1 = MixedRadixQuantity[{20, 32, 15.828}, {"AngularDegrees", "ArcMinutes", "ArcSeconds"}];
p2 = MixedRadixQuantity[{22, 32, 12.8}, {"AngularDegrees", "ArcMinutes", "ArcSeconds"}];

p1 - p2
(* Quantity[MixedMagnitude[{-1, -59, -56.972}], 
     MixedUnit[{"AngularDegrees", "ArcMinutes", "ArcSeconds"}]] *)

From the output above you can see the longer form using Quantity.

Hope this helps.

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    $\begingroup$ OK, I guess +1 for technical correctness, but...OMG...it's worse than COBOL. :) $\endgroup$ – Michael E2 Aug 27 '16 at 23:04
  • $\begingroup$ @MichaelE2 Of course a helper function can be used: dms=MixedRadixQuantity[#, {"AngularDegrees", "ArcMinutes", "ArcSeconds"}]& $\endgroup$ – Edmund Aug 27 '16 at 23:14
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    $\begingroup$ I suppose it's clear that I'm complaining about M, not your answer. I think carrying around these large, slow data structures internally is a waste. It's ok for UI purposes, although in this case, the formatted output is not highly readable. $\endgroup$ – Michael E2 Aug 27 '16 at 23:23

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