Is there a significance in ITF-style patterns, to which foot moves first or the direction in which it moves? Most of the patterns begin by moving to the left at the start. Thank-you!
I checked the encyclopedia, and found no reference to any symbolism of initially moving left or right. While there is significant symbolism for the number of movements, and the name of the hyung, there is nothing described about the left or right initial movement.
I also checked with some resources on Taekwondo’s influnces in Japanese kata and Chinese taolu. There, too, the line of the form (Japanese call it embusen, and the Koreans use the term yeonmuseon) has no purpose, other than for pure pedagogic reasons.
If there were any symbolic meanings, it would have been an afterthought by some instructor later on after the Encyclopedia had been written. That could have resulted from Choi himself having applied the concept later on, but I doubt that.
If you were asking from a purely technical reasoning - bunhae (bunkai in Japanese) then absolutely no. If your pattern has you turning left, you are responding to an attack which makes turning left advantageous; you could just as easily have turned right, instead, so long as you performed the entire form in the other direction, and that you assumed your attacker also performed his attack in a mirrored fashion. That is why a mirror performance of your hyung is a valid - if confusing - exercise, and should be encouraged, especially for those hyung who are not symmetric, and there are more than a few out there (Do-San, Hwa-Rang, Chung-Moo, and Se-Jong come to mind, I know there are others).
So, performing Do-San in a mirror has benefit, but not so much for Chon Ji - and definitely not Po-Eun. Chon-Ji is not perfectly symmetric, although it is close; Po-Eun is perfectly symmetric, and you would not realize any benefit from mirroring this hyung.
"Block with your left arm and strike with your right."
Most taekwondo patterns begin with a left arm technique (moving to the left) because all the patterns begin with blocks, since they are self defence routines. The only exception to this are the Sajus, which aren't really proper hyungs/tuls and have no historic background.
As for why blocking is associated with the left arm and striking with the right, the most common-sense answer is that for most people, the right arm is their dominant arm, which makes it feel more natural to strike on the right side.
Of course, this is mostly just a convention to look neat since realistically both arms should be equally capable of both blocking and striking.