The systematic study of inversion sequences avoiding triples of relations was initiated by Martinez and Savage. For a triple $(ρ_1,ρ_2,ρ_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced $\I_n(ρ_1,ρ_2,ρ_3)$ as the set of inversion sequences $e=e_1e_2\cdots e_n$ of length $n$ such that there are no indices $1\leq i<j<k\leq n$ with $e_i ρ_1 e_j$, $e_j ρ_2 e_k$ and $e_i ρ_3 e_k$. To solve a conjecture of Martinez and Savage, Lin constructed a bijection between $\I_n(\geq,\neq,>)$ and $\I_n(>,\neq,\geq)$ that preserves the distinct entries and further posed a symmetry conjecture of ascents on these two classes of restricted inversion sequences. Concerning Lin's symmetry conjecture, an algebraic proof using the kernel method was recently provided by Andrews and Chern, but a bijective proof still remains mysterious. The goal of this article is to establish bijectively both Lin's symmetry conjecture and the $γ$-positivity of the ascent polynomial on $\I_n(>,\neq,>)$. The latter result implies that the distribution of ascents on $\I_n(>,\neq,>)$ is symmetric and unimodal.