Let there be two identical particles with angular momentum operators
$J_1, J_2$ having the value $j$.
A state in the joined angular momentum basis will be $|J, m\rangle$
A state in the separated angular momentum basis will be $|m1, m2\rangle$
We shall observe the state with the highest azimuthal component $m = j + j = 2j$
|J=2j,m=2j\rangle = |m_1=j,m_2=j\rangle
It is easy to see that the state is symmetrical under interchange.
The $J_-$ operator is commuting with the permutation operator, hence every $|J=2j,m\rangle$
state is also symmetrical under interchange.
Next, we know that
|J=2j-1,m=2j-1\rangle = const * (|m_1=j,m_2=j-1\rangle - |m_1=j-1,m_2=j\rangle)
We can see that this next state with lower total angular is anti-symmetrical, and because of the previous statement every $|2j-1,m\rangle$ state is also symmetrical under interchange.
We can repeat this process down to the desired value of $J$