spherical projection of points to plane, precision error?

Working on a spherical to planar mapping and have run into what seems like a precision error and am not sure how to address. To start with, I'm considering two points on the sphere. I want to get a planar vector that represents the difference of these two points on the planar projection. To setup the projection, I assume the first point is the center of the projection and I create a rotation to bring this point onto the xz plane. I then project the second point onto the plane associated w/ the center point, then rotate this projection via the computed rotation. It does seem to work for the most part, but my accuracy seems to be only about 3 decimal places. If my math is correct, I should be able to compute this projection from p1 to p2 and from p2 to p1 and get the same answer but in reverse. Code and results are below. Anyone see what I'm doing wrong or see ways of improving the accuracy?

      static Quaternion ComputeRotation(Vector3 vector) {
         // compute xy angle/rotation
         var xyProjection = new Vector3(vector.x, vector.y, 0);
         var xyAngle = Vector3.Angle(xyProjection, Vector3.up);
         var xyRot = Quaternion.Euler(0, 0, (vector.x>0.0f) ? xyAngle : -xyAngle);
         // compute yz angle/rotation
         var yzProjection = xyRot * vector;
         var yzAngle = Vector3.Angle(yzProjection, Vector3.up);
         var yzRot = Quaternion.Euler((vector.z>0.0f) ? -yzAngle : yzAngle, 0, 0);
         return yzRot * xyRot;
     }

     public static Vector3 ProjectPoint(Vector3 centerVector, Vector3 point, float scale=1.0f) {
         var rotation = ComputeRotation(centerVector);
         var projection = rotation * Vector3.ProjectOnPlane(point*scale, centerVector.normalized);
         return projection;
     }

 [TestFixture]
 public class TestProjection {
     [Test]
     public void TestProjectPoint() {
         var p1 = new Vector3(0.195f, 0.981f, 0.0f);
         var p2 = new Vector3(0.377f, 0.911f, -0.167f);
         var pp12 = Projection.ProjectPoint(p1, p2, 1.0f);
         var pp21 = Projection.ProjectPoint(p2, p1, 1.0f);
         Debug.Log("pp12: " + pp12.ToString("F4") + " pp21: " + pp21.ToString("F4"));
         Assert.True(Geometry.CloseEnough(pp12, -pp21));
     }
 }

pp12: (0.1922, 0.0000, -0.1670) pp21: (-0.1949, 0.0000, 0.1638) UnityEngine.Debug:Log(Object) TestProjection:TestProjectPoint() (at Assets/Editor/Tests/TestProjection.cs:15)