It is well known that $9876543210/1234567890 = 109739369/13717421 = 8.0000000729...$
Recently I posted at
http://list.seqfan.eu/pipermail/seqfan/2012-October/010235.html
my observation that exactly the same ratio also could be expressed via another pair of , particularly as $7901234568 / 987654312 = 109739369/13717421 = 8.0000000729...$
and that as a result
$(7901234568 / 987654321) * 123456789 = 987654312$
which could be re-written as
$(7901234568 / 9876543210) * 1234567890 = 0987654312$
The above re-write (with slightly artificial but still valid pre-pending right hand side of the initial arithmetic identity with leftmost $0$) makes it to be the integer arithmetic identity, where all members are some specific permutations of all decimal base digits $1,2,...,8,9,0$ (with no duplicates)
So all four numbers in above expression (two pairs yielding same ratio) are base 10 permutations.
Is there only one such integer arithmetic identity involving permutations (with no duplicates) of all decimal base digits $1,2 \dots ,8,9,0$ (for base $10$) or there are more identities of the similar type?
The semi-brute force computational method to find the answer for this question is to construct list of all unique permutations (in ascending value order) and then to generate all possible products between elements, belonging to first half of the list (with smaller values), and those, belonging to the second half of the list (with bigger values), and then to look for such products, which are equal one to another in value.
What about other bases?