We propose a reproducible variant of the unblocked LU factorization for graphics processor units (GPUs). For this purpose, we provide Level-1/2 BLAS kernels that deliver correctly-rounded and reproducible results for the dot (inner) product, vector scaling, and the matrix-vector product. In addition, we draw a strategy to enhance the accuracy of the triangular solve via inexpensive iterative refinement. Following a bottom-up approach, we finally construct a reproducible implementation of the LU factorization for GPUs, which can easily accommodate partial pivoting for stability and be eventually integrated into a (blocked) high performance and stable algorithm for the LU factorization.