Introduction
What is "crouch canceling"? As a Samus main, I should probably be able to tell you. Upon finding the crouch cancel percent spreadsheet posted on reddit by user /u/cagliostro9, I quickly realized that my understanding of the phenomenon was based on intuition and hearsay and incomplete. Here is my attempt to clearly summarize the information I found from my research and tests.
ASDI Cancel
Why do we crouch cancel? I think the most common answer should be: "to retaliate," or, "to stay on the ground." Crouching, in fact, does not necessarily produce that outcome. What is actually responsible is automatic smash DI (ASDI). How it works is if you are holding the control stick or c-stick down on the last frame of hitlag (and you are close enough to the ground), the rest of hitstun is canceled. On the next frame the defender is placed on the ground in the landing animation, leaving you inactive for only 4 frames. This significantly reduces total stun in most cases, and often leaves you within range of your opponent on the ground.
This is generally known as "ASDI down," or "land canceling." For this post, I'll call this aspect of the crouch cancel phenomenon "ASDI canceling." It's important to mentally distinguish this mechanic from crouching, the reason being that you can be in any action state on the ground when hit and ASDI canceling still works.
Knockback
This is how the game determines how far you will be thrown after hit by an attack. Knockback is calculated based on a combination of factors including the attack's damage, the defender's weight, and the defender's current damage.
Crouch Armor
What effects does crouching have? When you hold down with the control stick, your character enters the squat animation. When hit in the squat animation (or squatwait), the knockback of the connecting attack is reduced by one-third, which in turn reduces stun. This may help us live longer, as well as sometimes provide a way to counterattack. When testing in 20XX, I noticed that you don't already need to be in squat when the attack connects: if you hold down on the exact frame the attack connects, you will receive the effects of crouch armor^1. But what does this mean in the context of ASDI canceling?
Crouch Cancel
ASDI canceling only works when the connecting attack does less than 80 knockback. Since crouch armor reduces knockback by one-third, ASDI down paired with crouch armor works until the connecting attack does 120 knockback. Because your character can suffer more knockback and still be able to land from ASDI, ASDI canceling works at higher percents for the defender with crouch armor than it would without.
I think this is what we generally mean when we talk about "crouch canceling," and what cagliostro9 called "true crouch cancel": holding down before an attack connects and then counterattacking with the reduced stun. But this is referring to two mechanics used in tandem. I can hold down on the frame the attack connects, release the control stick to neutral, and then hold down again on the last frame of hitstun (only two frames of input!), and I will receive the same effects as if I had been holding down the whole time, even since before the attack connected^2.
Crouching
You can enter the squat animation in the frame after an attack animation ends. Some attacks end with the attacker in squatwait.
When you return the control stick to neutral from crouching, you enter crouch recovery frames (squatrv). You can't reenter squat until after the eight frames of recovery. Beware! Crouch armor does not come into effect during crouch recovery, even if you are holding down. Curiously, you are able to walk out of crouch recovery, and you can crouch out of walk. This means you can return to squat in as soon as two frames instead of eight.
Sakurai Angle
A lot of attacks in this game have a special knockback trajectory. Whereas the majority of attacks always send the defender at the same trajectory angle, attacks with the Sakurai angle have a different trajectory angle depending on the knockback inflicted. If the attack does less than 32 knockback, the trajectory angle is zero (horizontal). If the attack does 32 or more knockback, the trajectory angle is 45 degrees.
Here's the kicker: ASDI canceling only works on attacks that have a vertical component to their trajectory angle. Attacks that do less knockback than the Sakurai angle threshold send the defender completely horizontal. Hitstun continues undeterred!
Since this mechanic depends on knockback, it is also affected by crouch armor. Crouch armor, when used, raises the knockback threshold to 48. In effect, crouch armor increases the knockback required for ASDI cancels to work at all. In some cases, this means you are decreasing knockback, but in turn receiving a net increase of stun as the defender. The misconception I had was that it if you wanted to "crouch cancel," it was always beneficial to crouch! But it turns out that in some cases if you want the reduced stun of an ASDI cancel, you must specifically not crouch.
It remains unclear to me how much there is to uncover with this other than perhaps a few edge cases. Attacks doing that little knockback have very little stun to begin with, so whether the defender suffers through it or not is usually inconsequential. Here is one situation where is seems to have some tangible effect. Say you have Fox and Samus waiting next to each other. Fox is at 71%. Samus's jab is a move with the special Sakurai angle. If Samus jabs Fox and Fox uses crouch armor, he cannot ASDI cancel and will only have time to shine. But, if Fox does not crouch before the hit, Fox is able to ASDI cancel and has enough time to grab Samus. This happens because the crouch armor in the first case put the knockback of the jab below the threshold required for ASDI canceling, even at 71%!
Conclusion
This post is by no means meant to present new, ground-breaking information, or anything that wasn't known before. Really it's just me starting to understand things that I could have known long ago by reading posts by Magus on Smashboards. If you haven't read those things and somehow got here first, I hope this helped you.
^1 Tested by freezing the game in developer mode. When an attack will connect on the following frame, hold down on the defender's controller and advance 1 frame. The control stick can be returned to neutral when the next frame is advanced.
^2 Tested by using Samus's f-smash on Bowser at 0%.
References
/u/cagliostro9's Fox crouch cancel spreadsheet
Doraki's guide on Smashboards
/u/V_D_X's 80KB percent spreadsheet
Magus posts
I KneeData Melee Calculator