Nice question! A comment about terminology. The object you are considering is a "balanced frequency square". Frequency squares generalise Latin squares in the sense that each row and column is a permutation of the same multiset of symbols. The term "balanced" means that each symbol occurs the same number of times.

I think these might be counterexamples:

112323

113232

232311

323211

231123

321132

or 

11223344

12142433

21241433

24433112

32131244

34412321

43314212

43324121


-Ian