Assume that $ (X,\mathcal{B},m,T) $ is a measure-preserving dynamical system, where $ (X,\mathcal{B},m) $ is a probability space, $ \mathcal{B} $ denotes all the measurable sets in $ X $, $ m $ is the measure on it and $ T $ is a measure-preserving map, i.e. for any $ B\in\mathcal{B} $, we have $ m(T^{-1}B)=m(B) $. If $ T $ is reversible, i.e. $ T^{-1} $ is also a measure-preserving map, show that $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^ix)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}f(T^{-i}x) $$ for $ m $-a.e. $ x\in X $.
In view of the famous Birkhoff theorem, the esxistence of limits for the RHS and LHS is ensured. However, I cannot show that the limits are the same. Can you give me some hints or references?
The answers using conditional expectations are right. But if you wish, you can see this fact using the Mean (Von Neumann) ergodic theorem. A good reference to this is the book of Einsiedler and Ward (your question correspond to the exercise 2.6.3). Let me explain some ideas to proof this. Let $(X, \mathcal{B},\mu)$ be a probability space and $T\colon X \to X $ a measurable automorphism. Consider $U_Tf := f\circ T $ the associated Koopman operator defined on $L^1(X,\mathcal{B},\mu)$. Note that $ U_T^j = U_{T^j}$. Let us denote the avarage of $f \in L^1$ under $T$ and $T^{-1}$ by $$A_nf := \frac{1}{n}\sum_{j=0}^{n-1}U^j_{T}f \quad \text{and} \quad A^{-1}_nf := \frac{1}{n}\sum_{j=0}^{n-1}U^j_{T^{-1}}f. $$ Note that we can restrict $U_T$ to the Hilbert space $L^2 \subset L^1$. The Mean ergodic theorem (see Theorem 2.2.1 in 1) tell us that for each $ f \in L^2(X,\mathcal{B},\mu)$, we have $$ A_nf \longrightarrow P_Tf \quad \text{and} \quad A^{-1}_nf \longrightarrow P_{T^{-1}}f \quad \text{as}\; n \to \infty \;\;\; \text{in}\; L^{2},$$
where $P_T\colon L^2 \to \mathcal{I}(T)$ denote the orthogonal projection on the closed subspace of invariant functions: $\mathcal{I}(T) := \ker(U_T - id) = \{ g \in L^2 : U_Tg = g\}$. Then, it is easy to see that $\mathcal{I}(T) = \mathcal{I}(T^{-1})$, thus $P_T = P_{T^{-1}}$. Notice that the conditional expectation as mentioned in the other answers is exactly $P_T$ in the case of $L^2$. We conclude that $A_nf$ and $A^{-1}_nf$ converge to the same function in $L^2$. Let us see that this is also true in $L^1$:
Corollary 2.2.2. (1): If $f \in L^1$, then there exists $f^* \in L^1$ such that $$\lim_{n\to\infty} A_nf = \lim_{n\to\infty}A^{-1}_nf = f^* \quad \text{in}\; L^1.$$
Steps to proof: Step 1: For each $f \in L^{\infty}$, we already know that there exists $f^* = P_Tf \in L^2$ such that $$ \lim A_nf = \lim A^{-1}_nf = f^* \quad \text{in}\; L^2.$$ Step 2: Prove that $f^* \in L^\infty$. Step 3: Using that $\|\cdot\|_1 \leq \|\cdot\|_2$, we can obtain that $$ \lim_{n\to\infty} A_nf = \lim_{n \to \infty} A^{-1}_nf = f^* \quad \text{in} \; L^1$$ Step 4: The corollary holds true for $L^\infty$ functions which is dense in $L^1$, proceed now by an approximation method to conclude to $L^1$.
Finally, note that Birkhoff theorem tell us that for every $f \in L^1$, there exists $f^* \in L^1$ such that $A_nf \to f^*$ almost everywhere and in $L^1$. So, $A^{-1}_nf$ must converge to the same function $f^*$ almost everywhere.