How well turbulence mixes a scalar $\theta$ is ultimately governed by the scalar dissipation rate $\chi =2D|\nabla \theta {|}^2$, where $D$ is the scalar diffusivity, making  scalar gradients central to turbulent mixing. We study the structure and amplification of these gradients for the canonical problem of passive scalar driven by a uniform mean-gradient in stationary isotropic turbulence, using well-resolved direct numerical simulations (DNS) at grid resolutions up to ${8192}^3$. The Taylor-scale Reynolds number $R{e}_{\lambda }$ spans $140-1000$, and the Schmidt number $Sc\equiv \nu /D$ spans $1-512$, where $\nu$ is the kinematic viscosity.  We analyze joint statistical correlations of velocity and scalar gradients that underlie scalar-gradient amplification. Unconditional statistics reaffirm earlier observations that production of scalar dissipation is dominated by nonlinear amplification of scalar gradients by strain-rate. Scalar gradients preferentially align with the most compressive strain eigenvector and remain orthogonal to vorticity, with  both trends virtually independent of $R{e}_{\lambda }$ and $Sc$. Conditional statistics reveal that this organization becomes dramatically enhanced in regions of intense scalar dissipation: scalar gradient becomes near-perfectly aligned with the most compressive strain eigendirection and orthogonal to the other strain eigendirections and to vorticity.  This geometry, also supported by visualizations, suggests that intense scalar dissipation is organized in sheet-like structures formed in shear layers between vortex tubes, where intense strain also generally resides. However, the effective strain acting along intense scalar gradients is comparatively much weaker, indicating intense scalar dissipation arises primarily from optimal alignments rather than intense strain alone.  Molecular diffusion arrests intense scalar-gradient events primarily by redistributing scalar-gradient variance away from intense structures. The contribution from imposed mean-gradient is negligible, but still imprints anisotropy directly onto smallest scales via the strain field. The conditional statistics broadly exhibit progressively weaker $Sc$-dependence as $R{e}_{\lambda }$ increases, demonstrating convergence towards an asymptotic state governing the local structure and dynamics of intense scalar dissipation.