Modular Arithmetic - two notations of congruence modulo

Question

I have found that if,

a ≡ b (mod m)

then

b ≡ a (mod m)

The meaning of a ≡ b (mod m) is that m | (a-b) & the meaning of b ≡ a (mod m) is that m | (b-a) but , (a-b) is opposite to (b-a) So , are we taking only the absolute value of the differences between a & b when we are dealing with congruence modulo?

Answer 1

$b \mid a$ means there's $c$ such that $a = b c$. Then $-a = b (-c)$, so that $b \mid -a$.

Answer 2

For any integers $r,s$, we have $r\mid s$ $\iff$ $r\mid-s$.

Answer 3

$a\equiv b\mod m$ can also be translated as

$a$ and $b$, divided by $c$, have the same remainder.

Hence the congruence relation is obviously reflexive, symmetric and transitive.