Rarity language in this game is easy to mix up because the texts use it in two different places.
Eggs have the five-name ladder
A recorded longer description says: steal Common, Rare, Epic, Legendary, and Mythic eggs. Those labels are on eggs. The same text does not point at a pet index that stamps the same tag on a hatched animal. Mythic is the top of that egg list, and the line "How high can you backflip, and can you collect them all?" is the collection challenge attached to climbing, not a drop rate.
Higher stages, in that wording, are where rarer eggs appear. Advancing is an access rule. It is not a stat bonus for owning a rare egg.
The live How to Play text does not repeat these five names. Check the egg you are looking at.
Pets are called rare, and some are mutated
The official hatch line is one sentence with two ideas: you hatch eggs to collect rare pets, and some of those pets are mutated. Mutation is an extra property on some hatched pets. It is not written as a sixth rarity tier, and it is not tied to income. The money line applies to your OP pets in general.
The description does not give mutation odds, a way to identify a mutated pet, mutation names, or an effect on stats, looks, or cash. It does not say you can add or remove a mutation after the hatch. Hatching is the only documented way a mutated pet enters the collection.
Update 2.5
On 29 September 2026 a public post referred to Update 2.5 for this game. A separate event note for that window said the event introduced a new mutation and still marked further details as unknown. No mutation name from that event is published in the sources read for this page. If you hatch after that date, read the pet's own labels instead of assuming the launch-era sentence still lists every mutation.
Keep egg tier and pet mutation as two columns in your own notes. A Mythic egg is not automatically a mutated pet, and a mutated pet is not automatically a higher egg tier.