= Solution
For the converse as well as the stated symmetric case, set
$$
a=\mathbb P_i(T_j<T_i),
\qquad
b=\mathbb P_j(T_i<T_j).
$$
Irreducibility and the existence of an invariant probability distribution make the chain positive recurrent, so $a,b>0$. During one return cycle from $i$ to $i$, the chain enters $j$ with probability $a$ and, after entry, makes a geometric number of visits to $j$ with success parameter $b$. Therefore
$$
\mathbb E_iN=\frac ab.
$$
On the other hand, the <stationary cycle occupation formula> gives
$$
\mathbb E_iN=\frac{\pi(j)}{\pi(i)}.
$$
Consequently
$$
\frac ab=\frac{\pi(j)}{\pi(i)}.
$$
It follows that
$$
\boxed{a=b\quad\Longleftrightarrow\quad\pi(i)=\pi(j)},
$$
which is precisely the claimed equivalence between symmetry of $i,j$ and equality of their invariant masses.
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