Solution:

Let the top row hold digits $a, b, c$ and the bottom row $d, e, f.$ In the sum of the two row numbers, the units digits satisfy $c + f \equiv 9 \pmod{10},$ and since $c + f \le 18$ in fact $c + f = 9$ with no carry. Repeating the argument in the tens and hundreds places gives $b + e = 9$ and $a + d = 9.$

The three column numbers add to $10(a + b + c) + (d + e + f) = 99.$ Writing $S = a + b + c,$ the bottom digits sum to $27 - S,$ so $10S + 27 - S = 99$ and $S = 8.$

Conversely, any digits with $a + b + c = 8$ determine the bottom row by $d = 9 - a,$ $e = 9 - b,$ $f = 9 - c,$ and both conditions hold. The number of solutions of $a + b + c = 8$ in nonnegative digits is $\binom{10}{2} = 45.$