Am I correct to assume that the possessive adjective has to match the gender of the noun?

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Hello David,I think you'll find that the possessive adjectives which change gender are nuestro/a, vuestro/a. Mi, Tu and Su do not change.  Saludos. John

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Hello David,When I copied and pasted my answer part of it dropped off. You may also be thinking of the long forms of Spanish possessive adjectives which would be "mío/a, tuyo/a, suyo/a. These also change with gender along with nuestro/a and vuestro/a. They are used when you want to emphasise possession e.g. "a friend of mine" as opposed to "my friend." There are of course plural forms which follow these gender changes.There is a lesson on this latter topic. If you search on "Long form possessive adjectives" you will find it.Saludos. John

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Hola David
As John explained and you can see in the table inside the lesson, only nuestro and vuestro have a differenciation for masculine and feminine. If you go back to the lesson, you will find a link to "long form possessive adjectives" that John mentioned. These do have different forms for masculine and feminine.
I hope it helps.
Saludos
Inma