If we reverse the digits of $12$ we will get $21$. $12^{2}=144$. If we reverse its digits we will get $441$ which is $21^{2}$.
Here is the puzzle. How many such two digit numbers are there? Digits must be different.
We got an algebraic method to solve it if the square is a three digit number.It given here :
Let $a>b$ $(10a+b)^{2}=100x+10y+z$ and $(10b+a)^{2}=100z+10y+x$
Difference between these two equations will lead to $a^{2}-b^{2}=x-z$
$x-z \leq 8$.Then $a^{2}-b^{2} \leq 8$.
Since squares are of three digits $a <4$.But since $a>b ,a \neq 1$.
$b=1 \Longrightarrow a^{2} \leq 9$.Then $a=1,2,3$ but only possibilities are $2 $ and $3$.
$b=2 \Longrightarrow a^{2} \leq 12 \Longrightarrow a= 1,2,3$. But possibility is $a=3$. So next such pair will be $13,31$. Higher values of $b$ will not work since $a$ cannot exceed 3. This is an explanation we obtained from a mathematics group.It works well if the square is a three digit number.What will happen if square is a 4 digit number... Or if we try to extend this problem for numbers having more than two digits ?
In response to your question: while there are many solutions where the squares have an odd number of digits, there are none for squares with an even number of digits such as 4.
Proof: Let $n \neq r$ be the number and its reverse, and let $10^{2k+1} < n^2 \neq r^2 < 10^{2k+2}$. Then $r^2$ ends with 1, 2, 5, 6 or 9 (not a 0, since then $n$ would start with 0). However, if $r^2$ ends with 1 then $n^2$ must start with 1, meaning $3.16 \times 10^{k} < n < 4.47 \times 10^{k}$; hence $n$ must start with either 3 or 4; but then $r^2$ ends with 9 or 6, a contradiction. By the same logic, if $r^2$ ends with 2 then $n$ starts with 4 or 5, meaning $r^2$ ends with 6 or 5. If $r^2$ ends with 5, $n$ starts with 7. If $r^2$ ends with 6, $n$ starts with 7 or 8. And if $r^2$ ends with 9, $n$ starts with 9. Since none of these are possible, no solution exists.
@tong_nor's answer demonstrates an infinite sequence of examples where the squares have an odd number of digits. In fact, there are many other such examples. A quick Python script suggests that all the the digits of the roots are always in the range 0 to 3. Proving this may be a good follow uo question.