Prior to the release of the album, "Dear Prudence" became the band's biggest hit in the UK, reaching No. 3 in September of the previous year. The song was intended to be a stand-alone single in Europe and as it was not issued as a single in North America, it later was added to the American version of Hyæna.
It is the only studio album that guitarist Robert Smith of The Cure composed and recorded with Siouxsie and the Banshees.
The album received both positive reviews and mixed reviews upon release. Melody Maker wrote:
Parts of it are so wistfully carefree that it's impossible not to credit Robert Smith as the talisman – his irreverence seems to course through everything. "Take Me Back" is the Banshees rollicking like some primitive jazz combo drunk on the Good Lord's wine. On "Belladonna", Smith's liquid guitar relaxes Sioux to the extent that she drops a few masks to reveal her vulnerability. When the siren sings "daylight devours your unguarded hours", she's illuminating her own predicament so acutely it surely can't be coincidence. "Dazzle", too, is naively daring: Siouxsie's voice, framed alone against the firmament of strings. It could be Lloyd Webber's Cats or something by Vaughn Williams. You can get impressed, wrapped up and lost in this.
Retrospectively, Stephen Cook gave a 4.5 star rating to Hyæna and wrote:
The emphasis here is on layered arrangements and pop tunes disguised as art-house production numbers ("Dazzle"); tasteful horn and keyboard parts expand the group's guitar-dominated sound and provide Siouxsie with an airy and dreamlike backdrop in which to fully display her considerable vocal talents.
When the album was reissued, The Quietus wrote a positive review and said:
"[it was] their most experimental work, Smith's presence in keenly felt on the disciplined execution of the grandiose "Dazzle" or the starkly seductive "Swimming Horses". But the real treasures were buried deep within the album. The lysergic Spaghetti Western twang of "Bring Me The Head of the Preacher Man" is evocative in its execution while the densely epic "Blow The House Down" finds Smith indelibly stamping his mark on the track courtesy of some his finest guitar work.