Cuyo is a relative pronoun with possessive value that agrees in gender and number with what is possessed, that which is after it in the sentence. The antecedent of the pronoun cuyo is the possessor of that which follows the pronoun.
The pronoun cuyo cannot agree with its antecedent because its adjective value forces it to agree with the noun that is after it.
I have a house whose (cuyo) balcony overlooks the sea.
Tengo una casa cuyo balcón tiene vista al mar.
My mother, whose (cuyos) eyes were green, had to wear dark glasses.
Mi madre, cuyos ojos eran verdes, tenía que usar lentes oscuros.
My sister, whose (cuya) appearance is beautiful, does not have to envy any model.
Mi hermana, cuya apariencia es hermosa, no tiene que envidiarle a ninguna modelo.
It is said that people whose (cuya) handwriting is ugly tend to be very intelligent and think very fast.
Se dice que las personas cuya letra es fea tienden a ser muy inteligentes y pensar muy rápido.