  We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given
Latin square that contain exactly $i$ different symbols.
  Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column
$j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals
in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv
t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the
number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac}
+ t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$.
  In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$
when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid
c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d)
\equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd
order with all row and columns sums equal to $k\equiv2\bmod4$.
