#Edit

Edit

In my opinion, what you want to do is more easily done when the button graphics and the model graphics are kept in separate lists. Also, your code can be simplified.

#Update

I think this code does what the OP is asking for. I have cut away a lot weeds from the OP's original code.

#Edit

In my opinion, what is asked for is more easily done when the button graphics and the model graphics are kept in separate lists.

In my opinion, what you want to do is more easily done when the button graphics and displayed graphics are kept in separate lists. Also, your code can be simplified.

Here is my take on improving your code. A lot more could be done to improve it, but the following code is reworked enough to accomplish (what I think is) your goal.

Manipulate[
  Column[{
    Row[
      MapThread[
        If[FreeQ[u, #2],
          Button[#1, AppendTo[u, #2]],
          Button[#1, u = DeleteCases[u, #2]]] &, 
        {solar, models}]],
    Row[u]}],
  {u, , None},
  {solar, None},
  {models, None},
  Initialization :> (
     solar = 
       {Graphics3D[{Blue, Sphere[{0, 0, 0}, 0.2]}, Boxed -> False],
        Graphics3D[{Red, Sphere[{0, 0, 0}, 0.2]}, Boxed -> False],
        Graphics3D[{Green, Sphere[{0, 0, 0}, 0.2]}, Boxed -> False],
        Graphics3D[{Orange, Sphere[{0, 0, 0}, 0.2]}, Boxed -> False]};
     models =
       With[{w = 108},
         {Pane[
            SphericalPlot3D[1 + 2 Cos[2 θ], {θ, 0, π}, {ϕ, 0, 2 π}, 
              PlotStyle -> Blue, Mesh -> None, Boxed -> False, Axes -> False],
            ImageSize -> w],
          Pane[
            SphericalPlot3D[Evaluate @ Abs @ SphericalHarmonicY[3, 1, θ, ϕ], 
              {θ, 0, π}, {ϕ, 0, 2 π}, 
              PlotStyle -> Red, Mesh -> None, Boxed -> False, Axes -> False],
            ImageSize -> w],
          Pane[
            ParametricPlot3D[{
              Cos[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
              Sin[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
              Sin[u/2] Sin[v] + Cos[u/2] Sin[2 v]}, 
              {u, 0, 2 Pi}, {v, 0, 2 Pi}, 
              PlotStyle -> FaceForm[Green, Green], Mesh -> None, 
              Boxed -> False, Axes -> False],
            ImageSize -> w],
          Pane[
            RevolutionPlot3D[{2 + Cos[t], Sin[t]}, {t, 0, 2 Pi}, 
              PlotStyle -> Orange, Mesh -> None, Boxed -> False, 
              Axes -> False],
            ImageSize -> w]}];
     u = models),
  TrackedSymbols :> {u},
  SaveDefinitions -> True]

#Update

I think this code does what the OP is asking for.

Manipulate[
  Column[{
    Graphics3D[
      MapIndexed[
        If[FreeQ[u, models[[#2[[1]]]]], 
          Button[#[[1]], AppendTo[u, models[[#2[[1]]]]]], 
          Button[#[[1]], u = DeleteCases[u, models[[#2[[1]]]]]]] &, 
        solar],
      ImageSize -> 300, Boxed -> False],
    Graphics3D[u,
      ImageSize -> 300, Boxed -> False]}],
  {{u, {}}, ControlType -> None},
  Initialization :> (
    solar = {
      Graphics3D[{Blue, Sphere[{0.5, 0., 0.}, 0.2]}, Boxed -> False], 
      Graphics3D[{Red, Sphere[{1., 0., 0.}, 0.2]}, Boxed -> False],
      Graphics3D[{Green, Sphere[{1.5, 0., 0.}, 0.2]}, Boxed -> False], 
      Graphics3D[{Orange, Sphere[{2., 0., 0.}, 0.2]}, Boxed -> False]};
    models =
     With[{w = 108},
       MapThread[
         Translate[#1[[1]], {#2, 0, 0}] &, 
         {{SphericalPlot3D[1.2 (1 + 2 Cos[2 θ]), {θ, 0, π}, {ϕ, 0, 2 π}, 
             PlotStyle -> Blue, Mesh -> None, Boxed -> False, 
             Axes -> False, ImageSize -> w],
            SphericalPlot3D[8. Abs@SphericalHarmonicY[3, 1, θ, ϕ], {θ, 0, π}, {ϕ, 0, 2 π},
             PlotStyle -> Red, Mesh -> None, Boxed -> False, 
             Axes -> False, ImageSize -> w],
           ParametricPlot3D[{
             Cos[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
             Sin[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
             Sin[u/2] Sin[v] + Cos[u/2] Sin[2 v]}, 
             {u, 0, 2 Pi}, {v, 0, 2 Pi}, 
             PlotStyle -> FaceForm[Green, Green], Mesh -> None, 
             Boxed -> False, Axes -> False, ImageSize -> w],
           RevolutionPlot3D[{2 + Cos[t], Sin[t]}, {t, 0, 2 Pi}, 
             PlotStyle -> Orange, Mesh -> None, Boxed -> False, 
             Axes -> False, ImageSize -> w]}, 
          10 Range[0, Length[solar] - 1]}]]),
  SaveDefinitions -> True]

And, yes, you can rotate each row of objects and the top row acts as buttons to show or hide the objects in the bottom row. The code is a bit rough in places, but I don't have time now to clean it up.

In my opinion, what you want to do is more easily done when the button graphics and the model graphics are kept in separate lists. Also, your code can be simplified.

#Update

I think this code does what the OP is asking for. I have cut away a lot weeds from the OP's original code.

Manipulate[
  Column[{
    Graphics3D[
      MapIndexed[
        If[FreeQ[u, models[[#2[[1]]]]], 
          Button[#[[1]], AppendTo[u, models[[#2[[1]]]]]], 
          Button[#[[1]], u = DeleteCases[u, models[[#2[[1]]]]]]] &, 
        solar],
      ImageSize -> 300, Boxed -> False],
    Graphics3D[u,
      ImageSize -> 300, Boxed -> False]}],
  {{u, {}}, None},
  {solar, None},
  {models, None},
  Initialization :> (
    solar = 
      {Graphics3D[{Blue, Sphere[{0.5, 0., 0.}, 0.2]}], 
       Graphics3D[{Red, Sphere[{1., 0., 0.}, 0.2]}],
       Graphics3D[{Green, Sphere[{1.5, 0., 0.}, 0.2]}], 
       Graphics3D[{Orange, Sphere[{2., 0., 0.}, 0.2]}]};
    models =
      MapThread[
        Translate[#1[[1]], {#2, 0, 0}] &, 
        {{SphericalPlot3D[1.2 (1 + 2 Cos[2 θ]), {θ, 0, π}, {ϕ, 0, 2 π}, 
            PlotStyle -> Blue, Mesh -> None],
          SphericalPlot3D[8. Abs @ SphericalHarmonicY[3, 1, θ, ϕ],  
            {θ, 0, π}, {ϕ, 0, 2 π},
            PlotStyle -> Red, Mesh -> None],
          ParametricPlot3D[{
            Cos[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
            Sin[u] (3 + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]), 
            Sin[u/2] Sin[v] + Cos[u/2] Sin[2 v]}, 
            {u, 0, 2 Pi}, {v, 0, 2 Pi}, 
            PlotStyle -> Green, Mesh -> None],
          RevolutionPlot3D[{2 + Cos[t], Sin[t]}, {t, 0, 2 Pi}, 
            PlotStyle -> Orange, Mesh -> None]}, 
         10 Range[0, Length[solar] - 1]}]),
  TrackedSymbols :> {u}]![demo][2]

And, yes, you can rotate each row of objects and the top row acts as buttons to show or hide the objects in the bottom row. Here is a rotated view.