Flow Graphs: Why do you need the symmetry property of a graph? $$\begin{gather}
f(u,v) \le c(u,v) \tag{Capacity constraint} \\
f(u,v) = -f(v,u) \tag{Symmetry} \\
\sum_{\large{v \in V, v \ne s,t}} f(u,v) = 0 \tag{Conservation of flow}
\end{gather}$$
When you are given a flow graph, it is commonly assumed it has 3 properties which are shown above.  Why is the property of symmetry required and not contained in the conservation of flow?  For instance, if the flow must sum to zero for a set of graph nodes, doesn't the flow in have to equal the flow out for each node thus symmetry isn't a necessary condition?
 A: A side remark: it is called a skew-symmetry, because of the minus sign (source). Also, your third rule should be
$$\sum_{v \in V} f(u, v) = 0, \quad \text{for all $u \in V \setminus \{s,t\}$},$$
because the flow between each vertex and start/target one should also be considered in the sum (see my explanation of the conservation of flow a bit lower), but it should not be considered for vertices $u \in \{ s, t \}$. Nevertheless, I use $0$ for that flow, so my example will work with your definition as well.
So, let's take $G = (V, E)$, where $V = \{ s, v_1, v_2, v_3, t \}$ and each two vertices are connected. Define $f$:
$$\begin{align*}
f(v_1, v_2) &= f(v_2, v_1) = 1, \\
f(v_2, v_3) &= f(v_3, v_2) = -1, \\
f(v_1, v_3) &= -1, \\
f(v_3, v_1) &= 1, \\
f(s, v) &= f(t, v) = f(v, s) = f(v, t) = 0, \quad \text{for all $v \in V \setminus \{ s, t \}$.}
\end{align*}$$
Let us now check the sums for the conservation of flow:
$$\begin{align*}
\sum_{v \in V} f(v_1, v) = f(v_1, s) + f(v_1, v_2) + f(v_1, v_3) + f(v_1, t) = 0 + 1 + (-1) + 0 = 0, \\
\sum_{v \in V} f(v_2, v) = f(v_2, s) + f(v_2, v_1) + f(v_2, v_3) + f(v_2, t) = 0 + 1 + (-1) + 0 = 0, \\
\sum_{v \in V} f(v_3, v) = f(v_3, s) + f(v_3, v_1) + f(v_3, v_2) + f(v_3, t) = 0 + 1 + (-1) + 0 = 0,
\end{align*}$$
which means that the conservation is satisfied, but skew-symmetry is obviously not. (Technically, each sum also includes $f(v_i,v_i)$, but this is always zero, so I chose to ignore it)
The conservation of flow simply states that "whatever comes into a node (different from start and target) will also go out", i.e., nothing will be lost at the nodes.
The skew-symmetry talks about relations between each two nodes: "whatever goes from $u$ to $v$, also goes from $v$ to $u$ but with the value of the opposite sign". In other words "You give me $\$1$, so I give you $-\$1$". This does not follow from the assumption that everyone gives and receives the same amount, as the above example shows (nodes $v_1$ and $v_2$ both give each other $\$1$).
