Image-based 3D scene reconstruction, which transforms multi-view images into a structured 3D representation of the surrounding environment, is a common task across many modern applications. 3D Gaussian Splatting (3DGS) is a new paradigm to address this problem and offers considerable efficiency as compared to the previous methods. Motivated by this, and considering various benefits of mobile device deployment (data privacy, operating without internet connectivity, and potentially faster responses), this paper develops Texture3dgs, an optimized mapping of 3DGS for a mobile GPU. A critical challenge in this area turns out to be optimizing for the two-dimensional (2D) texture cache, which needs to be exploited for faster executions on mobile GPUs. As a sorting method dominates the computations in 3DGS on mobile platforms, the core of Texture3dgs is a novel sorting algorithm where the processing, data movement, and placement are highly optimized for 2D memory. The properties of this algorithm are analyzed in view of a cost model for the texture cache. In addition, we accelerate other steps of the 3DGS algorithm through improved variable layout design and other optimizations. End-to-end evaluation shows that Texture3dgs delivers up to 4.1$\times$ and 1.7$\times$ speedup for the sorting and overall 3D scene reconstruction, respectively -- while also reducing memory usage by up to 1.6$\times$ -- demonstrating the effectiveness of our design for efficient mobile 3D scene reconstruction.