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We are of a family of eleven
When we join or depart numbers less than a cent
Excepting those special ten
We turn odd to even and even to odd
Like a tiny miracle of god
However, in general our company results in swap
Of the original ones, an easy sum for a clever chap

Hint 1:

If you add 09 to those (...) set of numbers...you get swapped result, similarly adding 18,..and with the exceptions to this being,....

What are we and what are those exceptions ?

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Are they

Multiples of 11 and the exceptions are the first ten even multiples of 11.

My reasoning:

We are of a family of eleven

Taking it's literal meaning to be numbers which are multiple of 11

When we join or depart numbers less than a cent

This means if we add or subtract from numbers which are less than 100

Excepting those special ten

We turn odd to even and even to odd

If we add odd multiples of 11 to a odd number it becomes even and to even number it becomes odd. So the exceptions are the even multiples of 11.

Like a tiny miracle of god

However, in general our company results in swap

Of the original ones, an easy sum for a clever chap

Here the swap means a number 'ab' becomes 'ba'. As we clearly know 'ab'+'ba'=11n. We see that using the right multiple of 11 we can swap the number. For example, if our number is 23 then 23 + 32 = 55 = 11 * 5. So, in the company of multiples of 11 we can swap numbers.

This is just my working, although I might be far off.

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  • $\begingroup$ Well, in the near right track, but requiring corrections $\endgroup$
    – Mea Culpa Nay
    Commented Sep 11, 2019 at 13:05
  • $\begingroup$ @MeaCulpaNay Requiring corrections in explanations of this solution? Or the solution itself is wrong? Because I have something else that might fit. $\endgroup$
    – Rand al'Thor
    Commented Sep 11, 2019 at 18:00
  • $\begingroup$ @Rand... please post your solution, as CMSnoob's answer I cannot consider as the correct one. $\endgroup$
    – Mea Culpa Nay
    Commented Sep 11, 2019 at 23:39
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Similar to the previous answer, but different:

you are multiples of $9$.

We are of a family of eleven
When we join or depart numbers less than a cent

$9,18,27,36,45,54,63,72,81,90,99$ - these are the family of eleven numbers you get by multiplying $9$ with numbers less than ten.

Excepting those special ten
We turn odd to even and even to odd
Like a tiny miracle of god

If you multiply $9$ by something with an odd number of digits, you get something with an even number of digits, and vice versa - unless the number is less than $11...1$. Those aren't just ten exceptions though, so probably there's a better explanation for this clue.

However, in general our company results in swap
Of the original ones

Swapping the digits in any of $9,18,27,36,45,54,63,72,81,90,99$ keeps you within the same set.

an easy sum for a clever chap

Summing the digits of any multiple of $9$ gives another multiple of $9$.

I'm not sure about all of the clues, but this seems like it might fit?

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  • $\begingroup$ Yes, you are in the right track, @Rand al'Thor. First explanation is correct. Remaining ones require edit to accept this as a correct answer. $\endgroup$
    – Mea Culpa Nay
    Commented Sep 12, 2019 at 8:30
  • $\begingroup$ @MeaCulpaNay Even the very last spoilertag isn't correct? That was one of the giveaways for me, the digit sum property. $\endgroup$
    – Rand al'Thor
    Commented Sep 12, 2019 at 8:31
  • $\begingroup$ No, not. That line is just for dramatization/ poetic purposes. $\endgroup$
    – Mea Culpa Nay
    Commented Sep 12, 2019 at 8:40

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