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The pair $Z_n=(X_n,Y_n)$ is an irreducible positive recurrent product Markov chain with stationary distributionThe stationary cycle occupation formula says that, during one return cycle to a state $z$, the expected number of visits to a set $A$ is $\Pi (A)/\Pi (z)$. TakeThenThe initial and terminal states both have $Y=0$, so either convention for including the endpoints gives the same count. Thus

For the converse as well as the stated symmetric case, setIrreducibility 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 $b$. ThereforeOn the other hand, the stationary cycle occupation formula givesConsequentlyIt follows thatwhich is precisely the claimed equivalence between symmetry of $i,j$ and equality of their invariant masses.