public Geometry Rotate( 
   Geometry geometry,
   MapPoint origin,
   double rotationAngle
)

{
  MapPoint pt = MapPointBuilderEx.CreateMapPoint(1.0, 3.0);
  MapPoint rotatePt = MapPointBuilderEx.CreateMapPoint(3.0, 3.0);

  Geometry result = GeometryEngine.Instance.Rotate(pt, rotatePt, Math.PI / 2);
}
// result point is (3, 1)

{
  // rotate a polyline

  MapPoint fixedPt = MapPointBuilderEx.CreateMapPoint(3.0, 3.0);

  List<MapPoint> pts =
  [
    MapPointBuilderEx.CreateMapPoint(1.0, 1.0),
  MapPointBuilderEx.CreateMapPoint(1.0, 5.0),
  MapPointBuilderEx.CreateMapPoint(5, 5),
  MapPointBuilderEx.CreateMapPoint(5.0, 1.0),
];
  polyline = PolylineBuilderEx.CreatePolyline(pts);

  Polyline rotated = GeometryEngine.Instance.Rotate(polyline, fixedPt, Math.PI / 4) as Polyline;  // rotate 45 deg
}

{
  var oid = 1; //ObjectID of the annotation feature to modify
               // Note: QueuedTask is required to access the AnnotationFeature
  {
    //Don't use 'Shape'....Shape is the bounding box of the annotation text. This is NOT what you want...
    //
    //var insp = new Inspector();
    //insp.Load(annotationLayer, oid);
    //var shape = insp["SHAPE"] as Polygon;
    //...wrong shape...

    //Instead, we must get the TextGraphic from the anno feature.
    //The TextGraphic shape will be the anno baseline...
    QueryFilter qf = new QueryFilter()
    {
      WhereClause = "OBJECTID = 1"
    };

    //annotationLayer is ~your~ Annotation layer

    using var rowCursor = annotationLayer.Search(qf);
    if (rowCursor.MoveNext())
    {
      using (var annoFeature = rowCursor.Current as
      ArcGIS.Core.Data.Mapping.AnnotationFeature)
      {
        var graphic = annoFeature.GetGraphic();
        var textGraphic = graphic as CIMTextGraphic;
        var textLine = textGraphic.Shape as Polyline;
        // rotate the shape 90 degrees
        var origin = GeometryEngine.Instance.Centroid(textLine);
        Geometry rotatedPolyline = GeometryEngine.Instance.Rotate(textLine, origin, System.Math.PI / 2);
        //Move the line 5 "units" in the x and y direction
        //GeometryEngine.Instance.Move(textLine, 5, 5);

        EditOperation op = new EditOperation();
        op.Name = "Change annotation angle";
        op.Modify(annotationLayer, oid, rotatedPolyline);
        op.Execute();
      }
    }
  }
}