Per the comments, and as already answered, the TextChanged event is not getting raised when you set the Text property to the value it already has.
It’s not clear whether this is something you can safely rely upon. It is a sensible optimisation, and I would be very surprised if future versions of .NET Framework drop it, but I cannot speak for older versions, nor for third-party implementations (Mono).
To be absolutely safe, I would not use the Focused check you put in your question. I would do exactly what the Text setter does now.
private void realText_TextChanged(object sender, EventArgs e)
{
var newMirrorText = Mirror(realText.Text);
if (mirrorText.Text != newMirrorText)
mirrorText.Text = newMirrorText;
}
This has the same advantage of preventing infinite recursion, but plays more nicely with other code you may put in your form that changes the text as a result of some other event.