University's excluding display because it "advocates for a position that involves political and social controversy" would violate the First Amendment

Public universities, like other government property owners, generally have fairly broad authority to restrict speech on their property, if they do so in viewpoint-neutral ways. Government property is generally divided into five categories (yes, five, not the three you might be thinking of). In a "traditional public forum" - basically limited to sidewalks, parks, and streets - the government gets no extra power to restrict the content of speech, though it can impose reasonable content-neutral time, place, and manner restrictions that leave open ample alternative channels for speech. Occasionally, the government may open up other property for all speakers and all subjects, which makes it a "designated public forum," where the rules are the same as in a "traditional public forum."

But sometimes the government opens up property only for certain speakers or certain subjects (e.g., rooms in a university that are available for student groups when no class is meeting there); that usually makes it a "limited public forum." And often the government just lets people into some place, not so they can speak but so that they can do other things (e.g., the hallways of a government building, or the inside of an airport); that is a "nonpublic forum." In either limited public fora or nonpublic fora, the government may generally restrict speech only if the restriction is reasonable and viewpoint-neutral. Finally, property that the government itself uses to convey its own views can be "not a forum at all" (e.g., a bulletin board on which the government posts announcements, or the podium in a government building on which government speakers stand); in those places, the government can even discriminate based on viewpoint.

Courts have generally concluded that the internal park space and sidewalk space within a university is not a traditional public forum. (Sidewalk space on the streets around the university might be a traditional public forum, and so could sidewalk space on normal city streets that go through a university, if the university is one of those urban campuses in the middle of town; but a quad within a university generally would not be a traditional public forum.) Instead, it's a designated public forum if the university opens it up to public speech (which it needn't do), a limited public forum if the university opens it up just for student group speech (to give one example of a speaker restriction), and a nonpublic forum if the university doesn't open it up for speech but does open it up for people just to walk through on their way to school, to sit on when having lunch, and so on.

The result is that universities tend to be relatively free (as a matter of First Amendment law, whether or not one thinks this is sound as a matter of academic policy) to impose content-neutral restrictions on demonstrations and other such speech, as well as imposing viewpoint-neutral subject matter restrictions (e.g., no partisan political rallies). But universities generally may not impose viewpoint-based restrictions.

That's how things mostly played out in a federal district court case last week, in Students for Life USA v. Waldrop (S.D. Ala. Feb. 4, 2015). The court seemed open to allowing a considerable amount of viewpoint-neutral restriction on speech, for instance limiting it to particular areas on campus. But even when the university could restrict all displays in certain areas, or restrict them in a content-based but viewpoint-neutral way, restrictions on "controvers[ial]" displays would generally be unconstitutional:

"[E]ven in a non-public forum, the law is clearly established that the state cannot engage in viewpoint discrimination - that is, the government cannot discriminate in access to the forum on the basis of the government's opposition to the speaker's viewpoint." The plaintiff argues that Mitchell and Steadman engaged in content discrimination and viewpoint discrimination when, while the First Policy was in force, they denied the plaintiff permission to place a cemetery of innocents in the Perimeter. The plaintiff has evidence that permission was denied because the plaintiff "advocates for a position that involves political and social controversy." The Court agrees with the plaintiff that this e-mail constitutes evidence that Mitchell and Steadman denied permission due to the plaintiff's viewpoint ("position") on abortion (pro-life). Because it was clearly established in February 2014 that such viewpoint discrimination violates the First Amendment, Mitchell and Steadman cannot receive qualified immunity with regard to these denials.

And indeed many court of appeals decisions have agreed that restrictions on "controversial" speech are viewpoint-based, because what makes speech controversial is often the viewpoint. A restriction on all views on a controversial topic, such as abortion, might in principle be treated as viewpoint-neutral, if the government instituted the restriction for reasons other than hostility to a particular viewpoint on the topic. How these sorts of specific exclusions of topics, such as "no speech about abortion" or "no speech about the war" - as opposed to more general exclusions, such as "no electioneering" or "no advocacy in favor of or against legislation" or "no vulgarities" - would be treated is something of an unsettled matter. But in this case, the court concluded that the university might well be focusing on the controversial nature of the viewpoint (speech that "advocates for a [controversial] position"), and not just employing a flat rule banning all speech one way or the other on controversial subjects.

I should note that the e-mail asserted that the proposed placement of the display would lead observers to wrongly assume that the university was endorsing the speech:

As you know, your organization advocates for a position that involves political and social controversy. Placing the crosses in proximity to Shelby Hall carries with it an implication that the College of Engineering endorses that position.

The University believes in the right of freedom of expression and it is my understanding that the area near the student center is specifically designated for that purpose.

I have copied the Dean of Students, …, with this message. I suggest that you contact him regarding any appropriate space for this activity.

But the court's view seems to be that, if the university is worried about mistaken perceptions of endorsement, it should exclude student group speech generally - whether or not its viewpoint is controversial - and not just speech that "advocates for a position that involves political and social controversy."