At issue in Greece v. Galloway is whether such invocations pass constitutional muster, even when government officials are not purposefully proselytizing or discriminating.
Can a town council, for example, open its meetings with prayers invoking Jesus Christ, as happened repeatedly in the town of Greece, N.Y.?
"There's a whole lot at stake here," said Ira Lupu, a law professor at George Washington University who specializes in the First Amendment's religion clauses.
"This case is about first principles: whether the government of a town, acting through its town board, can advance a particular brand of Christianity or any other faith," Lupu said.
"The government can't tell people what to pray," said Mateer, who argues that invocations are constitutional as long as government leaders are neutral in selecting who may offer an opening prayer. Of course the prayers offered in Greece were Christian, he said. "The people in the community are Christians."
For those who side with the town of Greece, the justices' decision could mean the protection or erosion of a cherished religious right.
"Americans today should be free as the Founders were to pray," said David A. Cortman of the Alliance Defending Freedom, the Arizona-based Christian law firm that represents the town. "In fact, the Founders prayed while drafting our Constitution's Bill of Rights."
The town in upstate New York used to open its town council meetings with a moment of silence. But starting in 1999, a "chaplain of the month" was invited to offer a prayer.
Two-thirds of those prayers offered in the course of a decade invoked "Jesus" or "the Holy Spirit," and pastors also asked those present to pray with them and recite the Lord's Prayer.
Two women from the town -- Susan Galloway, who is Jewish, and Linda Stephens, an atheist -- sued with the help of Washington-based Americans United for Separation of Church and State.
Greece did invite a few non-Christians to convene the council meetings in 2008, but only after the women filed the lawsuit, which charged that the town promoted Christianity in violation of the First Amendment's Establishment Clause, which prohibits government-sponsored religion.
Galloway and Stephens lost in federal court in 2011 -- the court ruled that town leaders weren't trying to exclude non-Christians -- but won in 2012 at the 2nd U.S. Circuit Court of Appeals, which ruled that the town's approach to public prayer amounted to an endorsement of Christianity.
Now the Supreme Court takes the case, with the Obama administration backing the town -- a surprise to many court watchers who expected the Democratic administration to side with those who want less religion in the public square.
Some speculate that a White House already at odds with conservatives on religious liberty issues, such as the contraception mandate in the president's health plan, sees the case as an opportunity to appear more sympathetic to those who say the administration has marginalized religion in the U.S.
Another theory: The administration believes the town is likely to prevail and wants to argue the case so that the court will opt for a more narrow definition of permissible prayer.
The town's lawyers will contend that prayer at Greece's town council meetings fell within the limits set in the last Supreme Court case that dealt with legislative prayer: Marsh v. Chambers, in 1983.
In that decision, the high court noted that the history of legislative prayer stretches back to the Continental Congress, which, in 1774, decided to convene with a prayer. It ruled legislative prayers -- even sectarian ones -- constitutional, as long as they don't promote any religion or disparage any other.
But since the Marsh ruling, said Mateer, lower courts have issued many other Establishment Clause rulings, with some decisions at odds with others, and judges have relied upon a variety of tests to determine whether the Establishment Clause has been violated.
"Establishment Clause jurisprudence today is in shambles," Mateer said. "This case presents the opportunity to the court to provide clarity."
Several court watchers predict a 5-4 split on the question and say the justices could go either way, with Justice Anthony Kennedy casting the swing vote as he often does. Though the court heard arguments Wednesday, it will likely take months before a decision is announced.