It's easy to remember if you apply the operation to whole number line instead of trying to remember the cases.

There are only 2 cases either you truncate to zero or you floor to negative infinity.

The zero behavior breaks the symmetry to help remember.

    n * truncate(a/n) + rem(a,n) = a
    n * floor(a/n) + mod(a,n) = a

Now suppose that "a" is any real, a/n shared by both. however truncate and floor will do different things to the negatives.

    truncate "moves" numbers to zero
    ..............................................................
    >>.>>.>>.>>.>>.>>.>>.>>.>>.>>0<<.<<.<<.<<.<<.<<.<<.<<.<<.<<.<<

    while floor "moves" number to -inf
    ..............................................................
    <<.<<.<<.<<.<<.<<.<<.<<.<<.<<0<<.<<.<<.<<.<<.<<.<<.<<.<<.<<.<<

when we then multiply by n its clear the positive "a" is the same however for negative a floor < truncate and so mod is positive while rem is negative

    for a < 0 : floor (a/n) < truncate(a/n) unless n divides a then they are =

The other case is for negative n, but that just flips the division and multiplication twice which has the effect of changing floor to ceiling and truncate to zero be truncate away from zero which flips the mod sign, but leaves rem alone, because its symmetric about zero.