I'm referring to in particular the meaning and interplay of +lat_0, +lat_ts, and +k_0. From reading through John Synder's text and Gerald Evenden's libproj docs, +lat_0, along with +lon_0, establishes the origin of the projected space. +k_0 specifies a scaling factor that generally applies for stereographic projections, but in the special case of polar-stereographic projections, specifying a +lat_ts accomplishes the same thing. Indeed, the PROJ4 library ignores +k_0 if +lat_ts is present. The geometrical interpretation of lat_ts is directly analogous to standard parallels; it is the latitude at which the projection-plane intersects the Earth. I've seen specs where the scale factor is given, and others where the latitude-of-true scale is given; either one does the job, but calculating one from the other is anything but straightforward.
Now consider the geotiff mapping. If lat_0 is determined to be a polar value, +lat_ts gets (erroneously) mapped onto +lat_0, as seen in ogr_srs_proj4.cpp in GDAL:
and is reflected in GDAL's .../frmts/gtiff/libgeotiff/geo_normalize.c
panProjParmId = ProjNatOriginLatGeoKey;
panProjParmId = ProjNatOriginLongGeoKey;
panProjParmId = ProjScaleAtNatOriginGeoKey;
panProjParmId = ProjFalseEastingGeoKey;
panProjParmId = ProjFalseNorthingGeoKey;
Thus for our polar projection, the origin is no longer at the pole, any scaling intended via +lat_ts is now lost, and any scaling given via +k_0 will be based off the unintended, non-polar origin. According to Synder & Evenden, the following should be equivalent, but this geotiff mapping effectively guarantees they aren't:
+proj=ups +south /* Universal Polar stereographic, defined in PROJ4 */
+proj=stere +lat_0=-90 +k_0=.994 /* see Synder, pg 157 */
+proj=stere +lat_0=-90 +lat_ts=-81.11453 /* Synder, pg 157 */
It seems to me that the geotiff mapping requires carrying along both +lat_0 and +lat_ts (where +lat_ts defaults to +lat_0 if not specified), especially if coordinate transformations are ultimately being handed off to PROJ4.