How to add two labels to a point in ListPlot, and how to control the precise positions of those labels?

Question

ListPlot has the option to label points using Labeled or Callout. At least those are the two ways to do it of which I am aware of. A simple example:
ListPlot[{Labeled[{1, 2}, "(1,2)"], Callout[{2, 1}, "(2,1)"]}, PlotRange -> {{0, 2.5}, {0, 2.5}}]

Now two questions:

How can I add two labels to one point? Each label should thereby be (roughly) opposite to its counter. Drawing the same plot twice as a workaround doesn't work. It would recognise the points as being too dense to display their labels.

Is there a way to have precise control over the label positions in terms of ocation angle and distance from the point? The option I know of is to specify Callout[point,label,location], where location could be Above,Below,Before,After,Left,Right, or a pair thereof, such as {Above,Right}. But the latter seems bugged, and I would also appreciate a tighter control than just a grid with 45° angular resolution.

Here a graphical description of what I would like to do. I cannot add the code for this plot, because the curve in it is numerical.

Thanks for suggestions!

kglr · Answer

SeedRandom[1]
points = RandomPoint[Circle[{0, 0}, {2, 3}], 10];
centroid = RegionCentroid@BoundingRegion[points, "FastDisk"];

labels1 = IntegerName /@ Range[Length@points];
labels2 = IntegerName[#, "French"] & /@ Range[Length@points];

gap = .2;

labeleddata = Table[MapThread[Callout[#, #2, # + i[[1]] gap Normalize[# - centroid], 
        Appearance -> "Leader"] &]@{points, i[[2]]},
     {i, {{-1, labels1}, {1, labels2}}}];

ListPlot[labeleddata, 
  LabelStyle -> 16,
  PlotStyle -> Directive[Red, PointSize[Large]], 
  AspectRatio -> 3/2, 
  PlotRangeClipping -> False, 
  ImagePadding -> 60, ImageSize -> 600, 
  PlotRange -> {{-2, 2}, {-3, 3}}, 
  Prolog -> Circle[{0, 0}, {2, 3}]]

Bob Hanlon · Answer

You can nest Labeled and Callout individually or together. They can be individually positioned.
ListPlot[{Callout[Labeled[{1, 2}, "Label 1", Above], "Label 2", Below,
    Appearance -> None],
  Labeled[Callout[{2, 1}, "Label 1", Above, Appearance -> None], 
   "Label 2", Below],
  Labeled[Labeled[{.5, .5}, "Label 1", Above], "Label 2", Below],
  Callout[Callout[{1, 1}, "Label 1", Above, Appearance -> None], 
   "Label 2", Below, Appearance -> None]},
 PlotRange -> {{0, 2.5}, {0, 2.5}}]

You can also use Show to overlap two plots using either Labeled or Callout
Show[
 ListPlot[{
   Labeled[{1, 2}, "Label 1", Above],
   Callout[{2, 1}, "Label 1", Above]},
  PlotRange -> {{0, 2.5}, {0, 2.5}}],
 ListPlot[{
   Labeled[{1, 2}, "Label 2", Below],
   Callout[{2, 1}, "Label 2", Below]},
  PlotRange -> {{0, 2.5}, {0, 2.5}}]]

For more precise positioning you can use Text in an Epilog
ListPlot[{{1, 2}, {2, 1}},
 PlotRange -> {{0, 2.5}, {0, 2.5}},
 Epilog -> {
   Text["Label 1", {1, 2}, {1.5, 1.25}],
   Text["Label 2", {1, 2}, {-1.5, -1.25}],
   Text["Label 1", {2, 1}, {1.5, -1.25}],
   Text["Label 2", {2, 1}, {-1.5, 1.25}]}]

How to add two labels to a point in ListPlot, and how to control the precise positions of those labels?

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