Is there significance behind the spider visions in Enemy?

Two theories are offered by Slate writer Forrest Wickman in an article about the ending, and (more to your question) by actress Sarah Gadon in an interview with Hollywood Reporter.

Forrest Wickman (totalitarian angle):

...Ultimately it's a parable about what it’s like to live under a totalitarian state without knowing it. It’s an Invasion of the Body Snatchers movie in which you don’t even realize it’s an Invasion of the Body Snatchers movie until the end—until it’s too late for our hero. In this case, the body snatchers just happen to be giant spiders.

Sarah Gadon (female intimacy angle):

"I think the spider[s are] a sign of his fear of female intimacy; it's a physical manifestation of his fear of female intimacy."

(The rest of this response is personal theory and speculation.)

I think both theories are correct and work well together. I particularly like the fear of intimacy theory and would go further: Adam is an oppressed, subtley misogynistic character who is attracted to but also subconsciously repulsed by and resentful of women, failing to understand them. When he dreams/sees spiders it's an expression of his oblique view of a species (or in general, a world) he doesn't "get".

Also:

I believe after the first scene of the movie, the spider scenes are dreams: a woman with a spider head, a Dali-esque city spider and finally the wife-as-spider transformation. Each dream is Adam's, and each one is longer and less distinctly a dream -- it's harder to tell when one begins or ends. The more surreal Adam's life becomes, the less it matters whether he's dreaming at all. Further, while the last scene in editing is the wife-as-spider transformation, I believe the actual last scene of the narrative is the first scene of the movie. I believe we're seeing Adam going to the sex club with his doorman in a flash forward. He's a regular there now. He has completely assumed his clone's life, wears his clone's wedding ring and wristwatch, and as an escape attends these sex shows like his clone did... Adam is so deep in the surreal that neither he nor we can know if he's dreaming anymore. By now, it doesn't matter.

To me it seems clear that there isn't a specific interpretation, rather its meant to be a piece of a puzzle that each individual will understand in a different way depending how they react emotionally to the film.

Here is an excerpt from an interview with the director, Denis Villeneuve, that explains this is exactly what the image is meant to do.

Can you talk about the imagery of the spider and what that represented for you?

The spider is a very precise image. The Double is a very complex, yet very simple story that is expressed in a very complicated way. There were some elements in the book that took 45 pages to express, and I said to myself, I cannot have the luxury to take 45 minutes to express such an idea, I need one image, a strong image. I always love when filmmakers are trying to express ideas with images that are beyond words, and the spider was a perfect image I found.

What I love about this image is that I think it’s a very strong and poetic image, but I liked the fact that you can understand it with your own sensibility. You can understand it, but it’s an understanding from inside yourself, not from an intellectual point of view, but more from what you feel as you see the image. I like to leave the audience free of interpretation, because for me, the pleasure of Enemy is an enigma. It’s really a puzzle and it’s designed in a playful way to play with the audience. So if I give the key from the start, it’s a bit boring. I love to leave the audience free of their own interpretation, but all the keys to understand the image are in this story.