There's one conservative interpretation of that phrase that's quite clearly articulated, though. It's a very narrow definition. It has the advantage of being clear, and it also has a sense of symmetry, so that the one clause about Establishments is exactly, equally balanced with the other religion-related clause of the amendment, "prohibiting the free exercise thereof."
Kelly Boggs, a Louisiana Baptist and the editor of the Baptist Message, stated this succinct and strict reading this way: "The government cannot enact a law that says you must give allegiance to a specific religion. Neither can the state pass legislation that says you can't pursue a particular religion. That's it."
That is to say, the one clause bars the government from forbidding belief, the other forbids forcing it.
Supreme Court Justice Clarance Thomas takes this same position. He recently wrote that "the Clause only prohibits 'actual legal coercion.'"*
There are (to put it mildly) some problems with this.
As Boggs works out this position, and applies this narrow definition of "respecting an establishment" to the case of a nativity displayed on government property in Texas, governments can be as religious as they want, as long as they don't demand or disallow any religious ascriptions.
So: a county can display Jesus as infant or man, as long as it doesn't mean one has to believe in Jesus. Legislative sessions can be begun with sectarian prayers, so long as no one is forced to pray. Public schools could teach religious doctrines as truth up to the point that students are made to confess those truths. A government could even declare an "official religion," especially, one would think, if it were official in the sense of being the foundational moral code for that government and it's legal system, so long as people are free to join or not join that official religion as they see fit.
So: an establishment of religion is a governmentally respected establishment of religion only when there is force involved.
As it stands, a government-funded, sponsored and authorized declaration of the truth of the Incarnation does not violate the Establishment Clause, since "Those who are offended by the public displays of nativity scenes are welcome to ... just ignore them."
If this interpretation of the Establishment Clause is going to hold any water, we have to imagine it, first, as being a non-Christian religion that's established but not "respected" in this sense of requiring and forcing belief.
Would Boggs, would Thomas, would we be OK with a city or county or state declaring itself Hindu?
Say the hypothetical government celebrated Hindu holidays, opened all sessions with Hindu prayers, used public space and/or funds to make Hindu declarations. One would still be free not to be a Hindu and free to be something else, but the government itself would be understood to be Hindu, and if one didn't like it, tough. Ignore it.
Most likely, Boggs and others holding to this interpretation of the First Amendment would object to this on the same grounds that conservative Germans use to restrict the freedoms of adherents to minority religions such as Jehovah's Witnesses and Muslims, that of leitkultur, i.e., "dominant culture," "head culture," or, more loosely, "traditional culture." One can even imagine the Confederate-symbol-supporting slogan re-purposed from "heritage, not hate," to "heritage, not faith" (which has the odd side effect of emptying that faith of any religious content, but never mind).
Regardless, the question of the thought experiment is still valid. Would that be OK? Would that Hindu town or county being officially and practically Hindu but just not requiring Hinduism of any of its residents be an acceptable set-up, not violating the First Amendment.
I doubt it.
A second, simpler objection, is that, even if one accepts this narrow version of what "respecting an establishment of religion" means, aren't taxes a kind of force? Isn't one, when paying the sales, property or payroll taxes that go the space used for a creche or a cross or another religious declaration, the salaries of the public officials hosting prayers or even themselves praying, or any other act of religion in government, being forced to support it?
One can of course look away from the nativity scene on the courthouse lawn, but one can't opt out of paying the taxes that make that nativity scene possible. Not without risking prison, anyway.
Boggs and Thomas' position on this is not clear to me. Boggs approvingly quotes the late Supreme Court Justice Hugo Black, who wrote, "No tax in any amount, large or small, can be levied to support any religious activities or institutions," but doesn't make the connection from that objection to the objection to a publicly-displayed nativity scene.
A third objection, a third problem, the most basic one, actually, is that this interpretation seems like a very weird interpretation of the language of the Bill of Rights. It requires some very elaborate syntactical twists to take "respecting an establishment of religion" to mean forcing one to affirm or assent or ascribe to a religion.
There's really no context, now or historically, where that's what "respecting" means. The interpretation only really makes sense, it seems, if one assumes a necessary symmetry between "prohibiting the exercise" and "respecting." But that's a strain and a stretch.
While the interpretation is nice and clear, and strict and narrow, and would make legal application pretty simple, since it's rather simple to decide if belief has been forced or not, it seems one would have to stomp and stamp on the actual language of the Establishment Clause to make it fit the interpretation that allows for governments to promote and proclaim Christianity.
*Thomas even further restricts this by saying the amendment still only applies to the federal government, despite the Fourteenth Amendment's Due Process Clause so that states can at least in principle, if they want, make laws criminalizing unbelief and/or belief or beliefs. That, however, is another argument.