@emendes Okay, I've made several changes. Highlights:

1. All conditions have been recoded to be indexed by names. The names that I (provisionally) used are AllI, Nonlinear, Different, `#I-D:I`, `#I-D:J`, NoPairs, and Symm. I'm not totally pleased with the 4th and 5th of these. Can you suggest something else, perhaps names that won't need quotes? It's fine if you use technical language.
2. Thank you for code snippet with nIs and degree. I made them input parameters, and I removed parms, which is now constructed internally.
3. I give some more control over the bugs associated with compiled and\or multithreaded Iterators, I've added keyword parameters numtasks and nocompile to the sequential that I already had. The default value of numtasks is kernelopts(numcpus), which is I believe 36 on your Linux machine. So, please try numtasks=35 for the example that errored out. If it still errors out, try adding nocompile also. That'll be a little slower, but it'll give @Joe Riel more info for when he gets around to looking at this. (I had no trouble doing the 8-tuples on my puny machine, but it takes 90 minutes.)
4. Several variables have been given more logical names. Feel free to suggest other such changes. I do strongly prefer a short name with a comment over a longer name that is itself the comment.
5. I've added several more comments, and elaborated several existing comments. If there are areas for which you'd like more comments, please suggest.

Please attach that PDF to your Reply.

I am eager to code the 7th condition: symmetry.

I think that you should mark this Answer (the parent of this Reply) as the Best Answer.

Here is the new work:

Download Symmetries.mw

@Grigoriy Yashin Yes, trig is one of the simplification algorithms that combine knows. To apply all of the algorithms that it knows, just remove the word trig. See ?combine, which lists the names of the algorithms. You may also want to apply simplify after combine.

@vv Thanks. Now I understand option threadsafe. It's just a way to flag a procedure as threadsafe in case the kernel incorrectly suspects otherwise; it overrides that suspicion.

Google has a much superior search algorithm, which is applicable here because Maple help pages are Google indexed (since they're somewhat duplicated as regular web pages). 

It burns me up that MaplePrimes Answers are not properly searchable by Google! The best hit that you'll get is the Answers page of the user who wrote the Answer. Since I have about 4000 Answers, someone Google searching for one of my Answers only knows that they have a positive hit; they'd need to wade through all the Answers to find it.

I got your email about this problem, where you attached a worksheet, which I re-attach here.

No Maple code that I have ever written (or ever will write) is intended to be used by Maple's 2D Input. Considering the length of the code, I refuse to look for the error in the 2D Input. If you redo it in the 1D Input form (code in a reddish monospaced font) that I originally posted it in, then I'll look at it. 

Download 23.mw

@acer I'm sure that I was the original author of that code, but it looks like it's been put through the wringer a few times, extracting all comments and formatting, and incorrectly modified a bit, and converted to 2D Input and then back to plaintext.

@emendes 

Re: Homorbital: As I said, that one is my own definition. Perhaps the concept is defined elsewhere with a different name,

Re: counterparts: Don't think in terms of what is removed. Nothing is removed in this program that uses an Iterator. Rather, only items that are desired for the final output are included in the first place. That's why this program is memory conservative. The more conditions that you add, the more time it will use, but the less memory.

Re: arguments: Yes, I'll add those things as arguments.

You say that parms can change. How exactly? Will there always be two indices? Is the first index always $1..nIs?

Can you think of names for the 6 conditions? It would be useful to have key names to replace the 1-6 used as table indices.

Can you describe how the tables of sets for conditions 4-6 are created? It would be useful to create them programmatically from some higher concept if possible.

Here is a worksheet with all-new and faster code, including passing the arguments. I do not pass nIs because it can be derived from parms.

Symmetries.mw

Please test scenarios that gave errors before. I might have fixed some of them. I can't test because I don't have Linux or a machine with 36 processors.

@nm Maple does have a limited ability to translate some Mathematica. See ?MmaTranslator.

@crashoverwide Oops, I have any intermittent GUI bug that causes minus signs to show up as K, even on graph axes. I just automatically translate them in my mind; unfortunately that's how common this bug is. So I just realized that I posted such a graph above. So, here's the graph with regular minus signs.

Does anyone know anything about this bug. It happens on both my computers. Plus signs show as C, and there are many other weird transliterations.

@dingtianlidi Here's a bonus problem, since you seem interested. Using the same type of graphical analysis, we can convert the same integration region to polar coordinates. For this, consider a ray emanating from the origin. Through which curve does it enter the region? Through which curve does it exit the region?
 

Download Polar_conversion.mw

@Steve Roper It's safer to generalize that if-statement to 

if not [args]::list(numeric) then return 'procname'(args) fi

Thus it'll work correctly regardless of the procedure's parameter names.

You want to "simplify" the given irrational algebraic constant? It's irrational, so it doesn't get much simpler than it already is. It's equal to 5*2^(-59/6), but that's hardly any simpler. What the point? If you're solving a practical physics problem about dimensioned quantities (such as temperature), then exact answers (other than 0, infinity, etc.) are irrelevant. Just use evalf.

@vv Yes, it's an impressive program. I guess that the input inequalities must be converted to polynomial? If that's the case, you may be able to avoid most solve failures by using solve(..., parametric, real).

@JAMET Now verifying the identity is trivial by hand or with Maple:

is((z1+z2)^2/(z1*z2) = (z1/z2 + z2/z1 + 2));
             true

@JAMET Let z1 and z2 both equal 1. Then the left side is 2 and the right side is 4. They're not equal.

@JAMET I think that you transcribed something wrong: (z1+z2)^2/(z1+z2) = (z1+z2), obviously.