So you want to learn about Pryoi?
Well, the Pryoi aren't auspices, though I've heard that comparison made. Sure, there are parallels, but rough ones. Like wolf moon-signs, Pryoi are inherent and influence one's dispositions. But the similarities between auspices and Pryoi stop there I think. Let me explain why I believe this is the case, as you asked of me, and in the process try to describe what exactly Pryoi are.
As an outsider you're missing a lot of context, so I will be as detailed as I can. This essay of sorts has benefited me as well as you, prompting me to explore the Pryoi in-depth... to understand the meaning and nuances of something most of us Folk take for granted.
I do wonder why a Garou would want to know about Pryoi. Is it academic curiosity? An attempt to understand the ways of my people? A boon is a boon, regardless of your intentions, and I repay my debts; I don't intend to owe you or any of your people one instant longer than I must. I can't imagine anything I tell you will compromise my people, and those of my tribe and yours have long shared cultural ties and mutual respect. I hope you do not prove me wrong in trusting you, Uktena.
In any case, I will try to leave my biases out of my writing as much as I can. Though I don't imagine I will manage much in this regard.

Influence
Pryoi resists definition and defies classification. Their impact on our minds and spirits is both subtle and significant. Pryoi reflect our inner nature, and influence who we are in equal measure. Like many things in Bastet experience, Pryoi are beyond mere knowledge, they are knowing.
From what I understand, auspices determine the role you play among the wolf-people. Those born under the full moon possess great martial prowess, and lead the battles against foes. The gibbous moons sing songs and tell tales to inspire the pack. Half-moons enforce laws and resolve disputes within their packs. Crescent moons master spirit-binding magics and communicate with spirits your kind press into service. And the new moons are spies, scouts and assassins, their subtle approach complementing the brutish full moons... and thus the moon-circle is complete. This being the case, it would follow your kind must gather in packs to be fully effective. Since a teller of tales must have an audience, and a leader must have followers to lead, and a warrior left to his own devices would rage and slaughter... well, perhaps the last example is best not dwelt on. But all this certainly frames your kind's propensity for group thought and action. And I would assume a member of each auspice should be represented for the optimum balance, so that no roles overlap and no role goes unfulfilled. Multiple wolves, one mind, yes?
We Folk are more self-sufficient: for us, it is one Bastet, one mind. We don't gather much, and so are used to doing for ourselves. On any given day I am my own warrior, healer, spirit-talker or scout; I can't fall back on others when I'm alone, which is most of the time. So I must be a pack unto myself. So it goes for all catkind. This is why Pryoi don't push us into narrow roles, for Gaia in Her wisdom knows that would interfere with our flexibility. Our strengths reflect our own desires, and aren't chosen for us; a Sun Cat is no more or less likely to be a hunter (or healer, shaman, warrior, or whatever "role") than a Bastet of the Twilight or Night.
But Pryoi do influence how we express our inherent talents, as well as our dispositions. The nature of a warrior marked by Day is markedly different from his Twilight sister or brother in Night. Different Pryoi complement certain qualities more than others; those born at Night find it easier to move by darkness and inspire fear. This doesn't contradict what I said before -- a propensity isn't a predetermination or a defined role. The inherent advantages a given Pryoi offers a Bastet are both subtle and open-ended: a warrior and shaman both gain equally from the Twilight Pryoi, but apply its blessings in different ways.

Changing Pryoi
As twilight follows the day and night follows twilight, sometimes a Bastet changes her Pryoi. This is known as "cycling." This isn't something that happens often -- for most Cats, anyway -- but in my experience most of us seem to cycle a few times in our lives. Cycling usually isn't a conscious decision, or at least it wasn't for me; it's more a natural life process that we heed. Sometimes there is a trigger that provokes a Pryoi change: a life event or spiritual calling. Other times we simply wake up one day with the cycle stirring within us. Most of us return to our "natural" Pryoi, the one that marked us when we were born. But sometimes the shift is more permanent. It's difficult to make generalizations about this sort of thing, as Bastet are so individualistic; what is true of of one Cat's cycling won't apply to the next. I don't know of your kind Cycle your auspices in such a way.
This is my personal theory about Pryoi change. There is a core Pryoi to each of the Folk, which can change but usually remains constant. A Bastet can stray from her core Pryoi, for days or even years at a time, but she will eventually return to it because it most closely reflects her inner nature.
Cycling is a process, though, not something one does on a whim. You remember me writing that Pryoi aid us in subtle ways? Such qualities change with Pryoi -- when I Night cycled a few years ago, I lost my Twilight affinities but gained the benefits of the Night. But there are a few days during the transition, before we fully "settle into a Pryoi," in which we seem to gain no benefits at all. It would seem best to not cycle while not undertaking any crucial task or endeavor, but one doesn't plan such things -- cycles happen when they happen.

The Three Pryoi
But I suppose you want more specific information on the three Pryoi themselves. Very well. I feel it's better you listen to two other Bastet, for I can only speak from my perspective. First I'll send you what an associate, Chikrava, wrote of the Day Pryoi. Then I will tell you of my own Pryoi, that of Twilight.

Cycling A Bastet may only ever enjoy the benefits of a single Pryoi at a time, but has the option to cycle out one for another. There are no rolls, renunciations or rituals involved; it simply happens over the course of approximately three days. During the transition, the character gain no benefit from either the Pryoi from which he's changing or the one he's going to adopt; in effect, he has no Pryoi. However, Pryoi Flaws -- Curious, Goldburn and Nightbound -- persist through cycles and can't be gotten rid of (beyond buying off the Flaw with experience points, if the Storyteller allows it). After the third day or so the new Pryoi takes effect. A player can opt to cycle or not, and only rarely will not have control over this (the Chaotic Cycles Flaw would be one exception to this). A Pryoi Gift can only be learned by werecats following the appropriate Pryoi. One can swap out Pryoi and learn each Gift in turn if she desires (and some Cats have done this), though this isn't a very good way to ingratiate oneself to the spirits. Merits & Flaws Chaotic Cycles (2-pt. Flaw): Pryoi changes tend to happen fairly often and at inconvenient times. This effect isn't entirely within the control of the player, though Storytellers should take care not to abuse this Flaw... too much. Chaotic Cycles may be taken in conjunction with Mercurial, so the werecat doesn't spend so much time in "downtime" with no Pryoi benefits; however, it can't be taken with Rigid. Fully a third of Celican are afflicted with Chaotic Cycles. Mercurial (2-pt. Supernatural Merit): You find it easier to cycle your Pryoi: it only takes 24 hours for you to transition fully. This Merit may not be taken with Rigid. Another fairly common trait among the Celican. Rigid (1-pt. Supernatural Merit): You find it inordinately difficult to change Pryoi. It takes no less than a week for you to cycle fully and begin gaining benefits from your new Pryoi, and Willpower difficulties increase by 1 during transition. This Flaw may not be taken with Chaotic Cycles or Mercurial.