There's actually several layers at work here, and there's also, unfortunately, no one-size-fits-all answer. I'll attempt to outline the various parts while not being overly wordy. I expect to fail on the latter, in part because the issues involved are interwoven...
So the first thing is deity relations. These are two-sided, and in many cases, the opinion Deity A has of Deity B is not the same as Deity B has of Deity A. Deity A may be unfriendly to Deity B, but Deity B may be neutral/indifferent to Deity A. As such, a divinely-empowered character of Deity A will have a different experience when casting spells upon a follower of Deity B than the converse situation.
On deity relations, the Divine Relation cantrip is a cantrip for mechanical reasons only. A cleric of a given deity would not have to cast a spell to know his/her deity's opinion of another person. They can look, ask and get a response in terms of a feeling. That feeling translates to Enemy, Unfriendly, Neutral, Friendly or Ally. Divine Relation never informs anyone of the target's actual deity. So it is not a "social faux pas" to use this cantrip. In fact, it's there to be used so clerics have the tools they need to in order to RP deity relations in the correct way. There's a whole lot between never casting Divine Relation and casting it "willy-nilly". Use it when it seems appropriate.
Regarding the question of a cleric casting spells upon you vs. you casting them upon another...In my opinion, the primary onus is on the caster, not the recipient, though how the recipient reacts to those spells is another matter. Again, due to deity relations, the caster's deity may have a neutral opinion of your charcter's deity, but your deity may have a harsher opinion of the caster's deity. In such a case, the caster can cast without penalty. As for "accepting" the blessing, you are correct in that there's no way to mechanically refuse such things. In such a case, it's up to your RP to do so. But...should you be refusing through RP?
As it turns out, this is not a simple answer! To a degree, our divine spells are already coded to take deity relations into account. Some spells have a lesser effect upon people and a greater effect upon others. Some spells will affect the caster differently depending on the deity of the target. Raise Dead/Resurrection come to mind in this latter category. Arguably some of the most powerful, beneficial magic a deity has to offer, a cleric should generally not be restoring life to a follower of an enemy deity except under the most dire of situations. That said, there's exceptions even to that which can depend on dogma and other IC considerations.
Regarding the question of whether or not one should accept spells from clerics of deities that your own considers to be an enemy or unfriendly...this can be a matter of interpretation. Some may think that such "support" is distasteful and a mockery of one's own beliefs and favored dogma, while others may take the attitude of allowing another deity to strengthen oneself because the act ultimately makes one's own deity stronger or furthers the goals of said deity. These ultimately come down to matters of RP and what makes sense for the character, dogmas and such. While not exactly the same case as what you're asking, I have an arcane character who refrains from casting on known Voraxians because they do not trust any magic but Vorax's and they don't trust arcane magic. Likewise, I've had Voraxians get angry after having arcane spells cast upon them no matter how beneficial they may have been. So the question of whether to accept spells from "enemies" is not clearcut, and given our 28 deities and 56 variations on their interpersonal relationships and dogmatic pressures, there's no real specific way to answer that and have it apply to everyone.
To address some specifically-mentioned points...
The explanation I was given was that it "gives the enemy deity power over me," but it seems like if anything the deity would be trusting me with a certain amount of their power, not the other way around.
No, it doesn't give the enemy diety any power over you, but neither is the enemy deity trusting you with any amount of their power. If anything, they're trusting their clerics. From an administrative sort of perspective, if a cleric of a deity frequently casts beneficial spells upon followers of enemy deities, either without bothering to check into the relationship or simply not caring (or worse, doing so for OOC reasons), the deity will respond and reduce or eliminate the cleric's powers for a time until lessons are learned. So on the recipient side of things, there's not really a down side, other than what your character might feel about it. The primary responsibility here is on the part of the caster.
Also, if we're in the same party, we're working towards the same goal. It's not like I'm receiving buffs from a Toranite and using them to go burn down a schoolhouse, I'm using them to further our group's collective mission. Even if our deities are enemies, our characters are, if only temporarily, allies. Regardless of our motivations, it should be expected that allies would help each other.
This is kind of OOC thinking, and for the casting cleric, it is not good enough that the "party" has a common goal for said cleric to treat enemies as allies. This is kind of one of those things that makes playing "evil" difficult at times. The goal is to have fun of course, but more often than not, it doesn't make any sense for Aeridinites, Toranites, Mistites, Rofireinites and Pyrtechonites to travel together just because their players think it would be fun to go see about some platinum. This matters much less when the characters are not divinely-empowered, but when they are (Clerics, Paladins and Champions), the question that has to be asked is whether or not it serves one's own deity to travel in such company.
Consider the case of using Resurrection. An Aeridinite might be inclined to resurrect a follower of Pyrtechon, because life is important to Aeridinites, even the lives of enemies. At the same time, that Aeridinite may not be too happy about the taking of life on behalf of the party itself, so Resurrection may not be cast, because death is also an important part of the cycle of life. A Rofireinite, however, may take no pity on a dead Pyrtechonite and in fact should have to face some pretty grave threat before resurrecting such a character (and OOCly, both players involved should know and understand this...even talk about it). Incidentally, Raise/Resurrection situations are cases where Divine Relation is really important.
So, as it seems I have failed on not being wordy, I'll offer a couple things to potentially help work through these things.
Overall, the best too we all have to head off any unnecessary "hardships" are to talk to the other players, OOCly, and work it out ahead of time. It's no secret as to your character's affiliation to other players, should they want to look. Yes, it's metagaming, but the point I'm getting at is that you could tell a cleric's player what Divine Relation would return without giving away any other IC info. As for any hardships for one's own character, in part, these are the sorts of things that one has to accept, whether implicitly or explicitly, when playing certain class, deity and alignment combinations. In your case, Pyrtechon's dogma is not terribly compatible with most group dynamics, and your character's status as a Champion probably makes that even more true. How does everyone playing nice-nice for the completion of some party goal in a temporary alliance further Pyrtechon's message, dogma and goals? That's going to be a tough one in most cases, because Pyrtechon has a lot more enemies than allies and friends combined (15 vs 7)...which I'm sure you already know!
Circling back around to your original question regarding accepting spells/blessings from deities that are enemies of your own, there's no prohibition against it. As a Pyrtechonite, you might take a kind of perverse thrill in knowing one of the Destroyer's enemies is making you stronger, even if for a short time. They do not, and will not, give said deity any power over you, except for what power you allow (i.e. if you are somehow swayed, even a little, away from Pyrtechon's dogma because of personal gratitudes or emotional responses brought on by an enemy helping you. As a recipient, you are not subject to any particular requirements for having spells cast upon you. In fact, personally, I'd say you could have a little fun with it and react with distaste at a healing spell from a Toranite, for example.
On the caster's side of things, they do have certain responsibilities to at minimum check in some way or the other and RP appropriately after the fact. We've had situations where clerics are casting buffs and healing spells and even raise/res. spells on followers of enemy deities without even a single word of RP on the matter. We're clearly not saying people can never cast on enemies, but we'd appreciate some attempt to RP the situation when they do...and to not make it a habit.
I hope this helped, even a little.