This, but in JS: https://gist.github.com/agirault/c32ccdc6755698f720d876374bebfa92

orientationLabel.js

import vtkImageMapper from 'vtk.js/Sources/Rendering/Core/ImageMapper';
import * as vtkMath from 'vtk.js/Sources/Common/Core/Math';

const { SlicingMode } = vtkImageMapper;

const ScreenSide = {
  top: Symbol('top'),
  bottom: Symbol('bottom'),
  left: Symbol('left'),
  right: Symbol('right'),
  close: Symbol('close'),
  far: Symbol('far'),
};

function LPSCoordinatesOfScreenSide(screenSide) {
  switch (screenSide) {
    case ScreenSide.top: return [0, -1, 0];
    case ScreenSide.bottom: return [0, 1, 0];
    case ScreenSide.left: return [-1, 0, 0];
    case ScreenSide.right: return [1, 0, 0];
    case ScreenSide.close: return [0, 0, -1];
    case ScreenSide.far: return [0, 0, 1];
    default: return [];
  }
}

const Label = {
  left: Symbol('Left'),
  right: Symbol('Right'),
  posterior: Symbol('Posterior'),
  anterior: Symbol('Anterior'),
  superior: Symbol('Superior'),
  inferior: Symbol('Inferior'),
};

function LPSCoordinatesOfLabel(label) {
  switch (label) {
    case Label.left: return [1, 0, 0];
    case Label.right: return [-1, 0, 0];
    case Label.posterior: return [0, 1, 0];
    case Label.anterior: return [0, -1, 0];
    case Label.superior: return [0, 0, 1];
    case Label.inferior: return [0, 0, -1];
    default: return [];
  }
}

function labelFor(screenSide, mapper) {
  // Method for image space slicing (I, J, K)
  function labelForImageSpaceSlicing(mat3x3) {
    // Create a 3x3 matrix transformation from the image space
    // (IJK, JKI or KIJ) to world space (LPS)
    const dir9 = mapper.getInputData().getDirection();
    const dir3x3 = [
      [dir9[0], dir9[3], dir9[6]],
      [dir9[1], dir9[4], dir9[7]],
      [dir9[2], dir9[5], dir9[8]],
    ];
    vtkMath.multiply3x3_mat3(dir3x3, mat3x3, mat3x3);
    
    // Apply the transformation to the coordinate of the side
    let outVec = [0, 0, 0];
    const inVec = LPSCoordinatesOfScreenSide(screenSide);
    vtkMath.multiply3x3_vect3(mat3x3, inVec, outVec);
    
    // Restrict to the unit vectors
    outVec = outVec.map((val) => {
      const threshold = Math.sqrt(0.5);
      if (val < -threshold) {
        return -1;
      }
      if (val > threshold) {
        return 1;
      }
      return 0;
    });
    
    // Find matching label
    const labels = Object.values(Label);
    return labels.find((label) => vtkMath.areMatricesEqual(outVec, LPSCoordinatesOfLabel(label)));
  }

  switch (mapper.getSlicingMode()) {
    case SlicingMode.I: {
      // JKI -> IJK
      const mat3x3 = [[0, 1, 0], [0, 0, -1], [-1, 0, 0]];
      return labelForImageSpaceSlicing(mat3x3);
    }
    case SlicingMode.J: {
      // KIJ -> IJK
      const mat3x3 = [[1, 0, 0], [0, 0, -1], [0, 1, 0]];
      return labelForImageSpaceSlicing(mat3x3);
    }
    case SlicingMode.K: {
      // IJK -> IJK
      const mat3x3 = [[1, 0, 0], [0, 1, 0], [0, 0, 1]];
      return labelForImageSpaceSlicing(mat3x3);
    }
    case SlicingMode.X:
      switch (screenSide) {
        case ScreenSide.top: return Label.superior;
        case ScreenSide.bottom: return Label.inferior;
        case ScreenSide.left: return Label.anterior;
        case ScreenSide.right: return Label.posterior;
        case ScreenSide.close: return Label.left;
        case ScreenSide.far: return Label.right;
        default: return null;
      }
    case SlicingMode.Y:
      switch (side) {
        case ScreenSide.top: return Label.superior;
        case ScreenSide.bottom: return Label.inferior;
        case ScreenSide.left: return Label.right;
        case ScreenSide.right: return Label.left;
        case ScreenSide.close: return Label.anterior;
        case ScreenSide.far: return Label.posterior;
        default: return null;
      }
    case SlicingMode.Z:
      switch (screenSide) {
        case ScreenSide.top: return Label.anterior;
        case ScreenSide.bottom: return Label.posterior;
        case ScreenSide.left: return Label.right;
        case ScreenSide.right: return Label.left;
        case ScreenSide.close: return Label.inferior;
        case ScreenSide.far: return Label.superior;
        default: return null;
      }
    default: return null;
  }
}

export default {
  ScreenSide,
  labelFor,
};

Not sure to fully understand, but it seems that you know what you are doing :-)

As a side note, I did not realize that Side was for screen positions, maybe ScreenSide would be more meaningful...