Q. I have noticed that a few spells are parts of two opposing schools of magic. For example, the limited wish spell is both conjuration/summoning and invocation/evocation. Can specialist wizards from either school use such spells? That is, can conjurers or invokers use limited wish? How is it that a spell can be from two opposing schools anyway?

A. A specialist wizard can use any spell that falls within the schools allowed to him, even if the spell in question also falls under an opposing school. Certain effects, such as limited wishes, can be produced in more than one way; in this case, what is wished for can be plucked from some other place and delivered, or it can simply be created to order.
It's an interesting idea, and could have some intriguing unintended consequences. If I were running a game under this assumption, it would affect what a Limited Wish could do, based on who was casting it.

If an Evocation specialist asked for a magic sword, then there would be one more magic sword in the world. If a conjuror asked for it, then she would get it, but a sword would disappear somewhere else in the world. (The original owner might eventually seek out the conjuror who stole his sword.)

Presumably, a Limited Wish from an Evocation wizard would not be able to take the party to safety. (But it might create a magic item that could be used to Teleport.)

The Summoner could produce a scroll with any spell that already existed, but couldn't use it to create a new spell that nobody in the world knew yet.

I might use it to determine in what way the wish was limited. Something conjured might eventually be conjured back by its original owner. Something created might be less than (or different from) what was expected.