... these are 2 separate ways of describing the same [angle] ...

There are an infinite number of ways to describe any angle. I see nothing inherently wrong with representing -0° 1' 0" (or 0°, -1', 0" in a different notation) as  (-1, 59, 0) internally as long as this tuple (and all such "weird" tuples; there are many in the test sets) correctly reduce to whatever you define as a "standard" representation (and the aforementioned tuple does reduce to  (0, -1, 0) as shown below). In the end, it's all a matter of the constraints you choose to impose on, e.g., notation and internal representation, user input, user output, etc., etc. These are all programmer choices.

c:\@Work\Perl\monks\thechartist>perl -wMstrict -le
"use TrigCalc qw(reduce);
 ;;
 my @dms = reduce(-1, 59, 0);
 print qq{(@dms)};
"
(0 -1 0)
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That said, simplest is usually best, so my choice for the internal representation of this angle would be -60 arc-seconds (or maybe decimal degrees, or radians, or whatever).