Hi Troubadoura,
It is possible, at least, to make a guess, because each kitten has 2 traits per category, one coming from the dad and one coming from the mum.
If the visible trait is coming from the mum, then the kitten's hidden one will come from the dad and vice versa.
Let's take a kitty as an example :
Fur: Russian - Black
Eyes: Odyssey Carnival (Mysterious|Big)
Shade: Natural
Tail: Genesis
Ears: Mysterious
Whiskers: Silver (Mysterious)
Size: Normal
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MOM: Kanji
Fur: Burmese - Platinum
Eyes: Key Lime (Shape: Curious | Pupil: Big)
Shade: Natural
Tail: Genesis
Ears: Mysterious
Whiskers: Silver (Shape: Mysterious)
DAD: Bill
Fur: Russian - Black
Eyes: Platinum (Shape: Curious | Pupil: Big)
Shade: Natural
Tail: Frisky
Ears: Mysterious
Whiskers: Silver (Shape: Guitar)
He is showing the same fur as the dad : Russian Black, since the mum's visible fur is more recessive than Russian Black,
then we can be sure this kitten either hides Burmese Platinum OR whatever more recessive fur the mum is hiding.
Of course, this works alike for the other 8 traits.