People on the left and the right keep talking about Cruz’s unpleasant face. I’ve heard such discussions, I’ve read them, and you’ve probably heard and read them too. Perhaps you even agree that Cruz has a remarkably unpleasant face.
Oddly, fewer people are talking about Trump’s unpleasant face, but I think it’s the case that he’s got one.
But I don’t care much about either guy’s face. I have never chosen or rejected candidates on that basis; it seems bizarre to me. I’m not above commenting on looks; for example, I thought Rick Perry looked mighty sharp with his new glasses, but I didn’t support him politically on the basis of that. I don’t even think that most people choose or reject candidates on the basis of looks, but it does get factored in, and perhaps it matters more and more in this age of television, Facebook, and Twitter.
There have been many presidents whose looks I find odd, even off-putting in various ways: Nixon, Truman, LBJ, Carter (whose face seemed almost Alfred E. Neumanish to me). If I look at pictures of all the presidents, I find that the most handsome presidents were Franklin Pierce and Woodrow Wilson, FDR, JFK, and Reagan, and those first two are probably not winning any prizes from most people for anything else but their faces:


On the other hand, you’d have to look long and hard before you’d find someone with a face as unattractive as Lincoln, considered by most people one of our greatest presidents. We’re already familiar with his mien, and register it as “LINCOLN” with all the attendant associations such as “great,” but if that were not true we might indeed find his face repellent.
And John Adams looks unimpressive or worse—rather weak—if you just look at his face without knowing a thing about his actual history and accomplishments:

Look at a page that has all the presidents’ portraits on it, and you’ll find them a rather odd bunch, actually.
For that matter, think about Churchill. Really think about Churchill. If you didn’t already know who he was, you’d probably think: what a strange, melting sort of face! What an unimpressive, pudgy body!:

But as with Lincoln, all is forgiven with Churchill appearance-wise because we know of his eloquence, courage, and achievements. But his face itself when looked at objectively is quite strange, although it has a certain stubborn power.
There was no internet when Churchill or Lincoln were running for office. But there’s plenty of evidence that most people who saw Lincoln in the flesh at the time found him not just ugly, but surpassingly ugly, with some people (under the heading “extreme observations” at that link) making it clear that he was actually the ugliest man they ever saw, which I suppose is a sort of distinction. Lincoln himself often joked about his face, with this being typical:
After being called two-faced in a debate: “I leave it to my audience. If I had another face, do you think I’d wear this one?”
As for Cruz, the internet is rife with people writing about why his face unsettles them. Here’s one such piece:
I have rarely, if ever, seen a conventional smile from Senator Cruz. In a natural smile the corners of the mouth go up; these muscles we can control voluntarily as well. But muscles circling the eyes are strictly under involuntary control: they make the eyes narrow, forming crow’s feet at the outside corners. Even the Mona Lisa’s smile shows this. The eyes give away one’s game and help us tell forged from genuine smiles. Grandma may have told you to put on a happy face, but you can’t if it isn’t heartfelt.
No matter what the emotional coloring of Senator Cruz’s outward rhetoric is, his mouth typically tightens into the same straight line. If it deviates from this, then the corners of his mouth bend down, not upwards. The outside of his eyebrows bend down, too, when he emotes, something so atypical that it disturbs me. Typically a person’s eyebrows arch up…
Actually, I’ve seen plenty of eyebrows that turn down, as well as mouths, and it doesn’t really disturb me although it doesn’t attract me either. The bearers of these turned-down faces aren’t usually people running for office, though; for example, I found an actor named Kyle Chandler (never heard of him before) who definitely has down-sloping eyes and eyebrows:

And I’m not quite sure what to say about this picture of 20s movie star Clara Bow—except that, well, she bears a slight resemblance to Ted Cruz here:

Quite a few websites suggest that it’s Ted Cruz’s smile that’s the problem—or rather, his supposed lack of a Duchenne smile. And if you look at a lot of photos of Criuz smiling, his smile does often look rather tight. But it’s not unlike that of his fellow Texan LBJ, and I don’t recall anyone mentioning a problem with LBJ’s smile at the time. There’s also an odd resemblance to the tight little smile of actress Joan Cusack, who is considered rather quirky and charming.
What I really think is the problem is that Cruz’s face looks too much like the classic mask of tragedy, which has the same down-sloping eyes and mouth:

As I said, I’m neutral about Cruz’s face. The following is not one of the worst photos I could find of him—I think it’s actually one of the best—but it’s certainly the way he looks quite a bit of the time, and I see nothing wrong there:
