rotate[\[Phi]_, \[Theta]_, \[Psi]_] := {{Cos[\[Phi]] Cos[\[Psi]], \
-Cos[\[Phi]] Sin[\[Psi]],
   Sin[\[Phi]]}, {Cos[\[Psi]] Sin[\[Theta]] Sin[\[Phi]] +
    Cos[\[Theta]] Sin[\[Psi]],
   Cos[\[Theta]] Cos[\[Psi]] -
    Sin[\[Theta]] Sin[\[Phi]] Sin[\[Psi]], -Cos[\[Phi]] \
Sin[\[Theta]]}, {-Cos[\[Theta]] Cos[\[Psi]] Sin[\[Phi]] +
    Sin[\[Theta]] Sin[\[Psi]],
   Cos[\[Psi]] Sin[\[Theta]] + Cos[\[Theta]] Sin[\[Phi]] Sin[\[Psi]],
   Cos[\[Theta]] Cos[\[Phi]]}}

face[\[Theta]_, \[Phi]_, \[Psi]_, p2_, p3_, ori_] :=
 Module[{v1, v2, v3, v4},
  v1 =  rotate[\[Phi], \[Theta], \[Psi]].ori;
  v2 =  rotate[\[Phi], \[Theta], \[Psi]].(ori + p2);
  v3 = rotate[\[Phi], \[Theta], \[Psi]].(ori + p2 + p3);
  v4 = rotate[\[Phi], \[Theta], \[Psi]].(ori +  p3);
  Polygon[{{v1.{1, 0, 0}, v1.{0, 1, 0}},
    {v2.{1, 0, 0}, v2.{0, 1, 0}},
     {v3.{1, 0, 0}, v3.{0, 1, 0}},
     {v4.{1, 0, 0}, v4.{0, 1, 0}}}]
  ]

renderorder[\[Theta]_, \[Phi]_, \[Psi]_, p2_, p3_, ori_] :=
 Module[{v1, v2, v3, v4},
  v1 =  rotate[\[Phi], \[Theta], \[Psi]].ori;
  v2 =  rotate[\[Phi], \[Theta], \[Psi]].(ori + p2);
  v3 = rotate[\[Phi], \[Theta], \[Psi]].(ori + p2 + p3);
  v4 = rotate[\[Phi], \[Theta], \[Psi]].(ori +  p3);
  Mean[Flatten@{{v1.{0, 0, 1}},
     {v2.{0, 0, 1}},
      {v3.{0, 0, 1}},
      {v4.{0, 0, 1}}}]
  ]

shader[\[Theta]_, \[Phi]_, \[Psi]_, p2_, p3_, ori_] :=
 Module[{v1, v2, v3, v4, lvec, nvec},
  v1 =  rotate[\[Phi], \[Theta], \[Psi]].ori;
  v2 =  rotate[\[Phi], \[Theta], \[Psi]].(ori + p2);
  v3 = rotate[\[Phi], \[Theta], \[Psi]].(ori + p2 + p3);
  v4 = rotate[\[Phi], \[Theta], \[Psi]].(ori +  p3);

  lvec = {-1, 1, 1};

  nvec = Cross[v2 - v1, v3 - v1];

  1 - ArcCos[lvec.nvec/(Norm[lvec] Norm[nvec])]/\[Pi]

  ]

surfaces = {
   {{1, 0, 0}, {0, 1, 0}, {0, 0, 0}},
   {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}},
   {{1, 0, 0}, {0, 0, 1}, {0, 0, 0}},
   {{1, 0, 0}, {0, 0, 1}, {0, 1, 0}},
   {{0, 1, 0}, {0, 0, 1}, {0, 0, 0}},
   {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}};

Module[{\[Theta] = 0, \[Phi] = 0, \[Psi] = 0, \[Delta] = \[Pi]/24,
  order, shade},
 Framed@Pane[
   EventHandler[Dynamic[
     order =
      Transpose[
        SortBy[Table[{i,
            renderorder[\[Theta], \[Phi], \[Psi], surfaces[[i]][[1]],
             surfaces[[i]][[2]], surfaces[[i]][[3]]]}, {i, 1,
            Length[surfaces]}] // N, Last]][[1]];
     Show[
      Graphics[
       Table[{(GrayLevel[
              shader[\[Theta], \[Phi], \[Psi], surfaces[[#]][[1]],
               surfaces[[#]][[2]], surfaces[[#]][[3]]]] & /@
            order)[[i]], (face[\[Theta], \[Phi], \[Psi],
              surfaces[[#]][[1]], surfaces[[#]][[2]],
              surfaces[[#]][[3]]] & /@ order)[[i]]}, {i, 1,
         Length[surfaces]}]]
      , PlotRange -> {{-2, 2}, {-2,
         2}}]], {"UpArrowKeyDown" :> (\[Theta] = \[Theta] + \[Delta]),
      "LeftArrowKeyDown" :> (\[Phi] = \[Phi] - \[Delta]),
     "DownArrowKeyDown" :> (\[Theta] = \[Theta] - \[Delta]),
     "RightArrowKeyDown" :> (\[Phi] = \[Phi] + \[Delta]), {"KeyDown",
       "w"} :> (\[Psi] = \[Psi] - \[Delta])}]
   ]]