Question: I recently rewatched *The Prestige* and find this question to be central to understanding the plot. In the final narration Cutter seems to be challenging the audience to discover a further, hidden secret. This secret would necessarily not be made explicit in the movie, but nevertheless be key to understanding the entire plot.

I think it is possible that Angier did not know that Borden was backstage. Angier had planned all along to stage his death after 100 shows, so that he could return to being the Lord full-time. Perhaps he also realized that he could not go on murdering himself indefinitely, so he put a hard limit on the practice. Borden's presence backstage was therefore a coincidence.

Another possibility is that Angier did know Borden was there spying, because Fallon had told him. After Sarah's suicide, Fallon secretly wanted his twin brother dead. Possibly he involved Cutter in the scheme at some point. Then Fallon killed the Lord in order to retrieve his daughter, whom Angier did not realize was actually Fallon's.

Answer: 
I think Angier was prepared for, and was expecting Borden to crash his show, but he did not know when Borden would do it. Angier clearly did not have the courage and plan to keep killing a duplicate of himself forever. He wanted to outdo Borden mainly for revenge. It was his plan than sooner or later Borden would show up, and get himself incriminated for the accident, and land up in jail, thus, freeing Angier from ever having to do a magic show again. He could retire in peace as the 'greatest showman', revealing his secret to Borden at the jail.

But he was clearly not prepared to meet Borden's twin. And he is visibly surprised to find such a simple answer to Borden's illusion.
This is one of those things that could probably never be answered, I am not even sure the author really knows. All the comments above make good points, so not to distract from them, but felt this deserved at least one answer ...

The final narration and the 'clues' that are revealed are meant to be provocative. A set of identical men with different personalities and a man duplicating himself destroying each other (and themselves). So there is the commentary of self-destruction and, ambition and envy/greed destroying the things one values. The only way to end the cycle was for the driving forces to be stopped. The final narration and images/shots are deeply metaphorical.

So, I believe @Sean is correct above; Fallon was the calm one and wanted the destruction to end. With the nod from Cutter at the end we are lead to believe that there was an understanding between them. Cutter can be seen as having the same reservations (conflict) with the actions of Angier and Borden near the end. 

So, I don't believe Angier "knew" Borden was backstage, but took advantage of the opportunity (of which he may have desired to happen and planned for) and was not affected by the misery that followed, then paid for his own sins.

The movie gets better each time I watch it. I think it is very undervalued.