I'm an Air Force veteran and every year there is an air show/open house hosted by some base somewhere. It's a huge pain in the ass for everyone on the base because it gets open to the public and you're almost definitely "volunteering" to either augment base security, direct traffic, set up, clean up, etc.
There are a few reasons it's done, but one of them is to give people some sort of idea where their tax money is going. The sexy part of it anyway. We don't really show off our office supplies and grounds maintenance contracts to the public.
Were there any equivalents to this in any Ancient Cultures, specifically Rome? Obviously there's no ancient version of the Thunderbirds, but were there military demonstrations or major events geared toward public interaction? It seems like something like that would be used to reinforce public pride in the power of the Empire, Republic, whatever.
I don't know if anything like this existed in Rome, but the Athenian cavalry frequently put on shows, both to entertain the people and to show off its quality to government officials. Classical Athens paid 40 talents a year (a vast sum of money) for the upkeep of its cavalry, and, exactly as you say, the people needed to know that their money was being well spent.
Now, Classical Greece had no barracks or military bases, so the cavalry was displayed in public spaces. It took part in processions during religious festivals and carried out some public manoeuvres (including a javelin-throwing contest) at the Lyceum. But the most spectacular show was the review before the Council, given yearly at the city's chariot race track:
When the Hippodrome is the scene of the display, the right plan would be that the men should first be drawn up on a front broad enough to fill the Hippodrome with horses and drive out the people standing there. In the sham fight, when the phylai ["tribes", the 10 constituent units of the cavalry] pursue and run from one another at the gallop in two formations of five phylai, each side led by its commander, the phylai should ride through one another. How formidable they will look when they charge front to front; how imposing when, after sweeping across the Hippodrome, they stand facing one another again; how splendid, when the trumpet sounds and they charge once more at a quicker pace! After the halt, the trumpet should sound once more, and they should charge yet a third time at top speed; and when they have crossed, they should all range themselves in battle line preparatory to being dismissed, and ride up to the Council, just as you are accustomed to do.
-- Xenophon, The Cavalry Commander 3.10-12
The purpose of all this was exactly the same as that of the air shows you're describing - to provide a spectacle, inspire the citizens with awe, and make them proud of the armed forces that were being funded with their tax money. When the Athenians instituted a formal military training programme for the heavy infantry in the 330s BC, they also made its cadets parade and manoeuvre before the Council and the citizens to show off what they had learned. The participation of these cadets in torch processions would also have been an imposing sight.
All of this was a huge pain in the ass for the Athenian cavalry, too, because it meant they actually had to train. Greek militias were not keen on such hard work and generally didn't bother to practice their skills, but if the Council got to judge you, you pretty much had no choice. The actual review exercises were therefore usually very simple, requiring minimal coordination and practice. The bit of Xenophon I've cited above is actually not what really happened, but what Xenophon recommends should happen; note the constant use of "the right thing would be... they should..." and so on. He says elsewhere in the same treatise that only a commander with unusual influence could get his cavalry to perform some nice drills. It is clear from his account that the actual exercises were a lot less imposing than he would wish.