Here's the thing: because All The President's Men is based on cold, hard reality, there aren't any symbols in this film that didn't exist in the cold, hard, disco-infused 1970's. That's not to say that symbolism doesn't exist in the real world—some of our favorite literary symbols are grounded in what actually happened.
There really are white whales. Puritans really did punish adulterers by making them wear letters. And we know that caged birds actually do sing…although we need a little help with why they sing.
And there are a few recurring images that give All The President's Men a strong narrative cohesion. The first is telephones.
In today's era of Snapchat and WhatsApp using a landline, complete with curly phone cord, is downright archaic. But it's the way people had to communicate in the 70's, and it emphasizes a general lack of communication. People don't want to talk to Woodward and Bernstein anyway. The phone gives them yet another option to not talk. They can slam doors in their faces, and they can hang up on them.
But what could be more boring than watching someone talk on the phone? All the President's Men manages to make this act interesting. Alex Woo, a blogger and story artist at Pixar, has some awesome analysis focusing on critical scene involving a dual-focus lens and Woodward on the phone with a source. (Source)
The camera pans in, showing us Woodward's laser-sharp focus on his job. And he's often the only person on the phone in many of his scenes—you can see phones going unused in the background. This illustrates that Woodward is determined where many of his colleagues are not. Today, texting on the job is a big problem. But in Woodward's day, a man could actually get ahead by being on the phone 24/7.