Romans 1:24-27 is an ugly text. I don't like to quote it, but it's important to do so:
Therefore, God handed them over to impurity through the lusts of their hearts for the mutual degradation of their bodies. They exchanged the truth of God for a lie and revered and worshiped the creature rather than the creator [...] Therefore, God handed them over to degrading passions. Their females exchanged natural relations for unnatural, and the males likewise gave up natural relations with females and burned with lust for one another. Males did shameful things with males and thus received in their own persons the due penalty for their perversity.The most important thing to notice about it off the bat is that it's taken out of context. I said at the beginning that all these passages are about something other than homosexuality; when you look at the context, this passage is primarily about idolatry, and the homosexual behavior is secondary to the idolatry. "Secondary" not only numerically but also in a medical sense, which is to say "symptomatic of."
The real key to understanding this text are the words "unnatural" and "perverse." The words "natural" and "nature" come from the same root as the words "native" and "nativity," meaning birth. Therefore, one's "natural" state or condition is the state or condition one was in at birth. "Perverse" means turning away from.
In Paul's time, people probably believed that everyone was born inherently heterosexual. It's safe to say that some people believe that now, but that doesn't mean they're right. There's no scientific evidence to support that. There IS, however, evidence to suggest that there's a genetic predisposition towards homosexuality, and if it's a genetic predisposition, it's entirely possible that sexual orientation is congenital, meaning existing from birth. And if it exists from birth, that means that it's a person's natural condition.
In this text, Paul is saying that it is sinful for people with a natural heterosexual orientation to "turn away" from their natural state and perform homosexual behaviors. He's not saying anything about people with a natural homosexual orientation, but the obvious logical conclusion is that it would be just as bad for them to "turn away" from their natural state and perform heterosexual behaviors. In either case, the parties involved would be enacting a lie.