diff --git a/tests/BeamCenterFromBackgroundTest.cpp b/tests/BeamCenterFromBackgroundTest.cpp
index dc0007b90..af82090c7 100644
--- a/tests/BeamCenterFromBackgroundTest.cpp
+++ b/tests/BeamCenterFromBackgroundTest.cpp
@@ -124,3 +124,24 @@ TEST_CASE("BeamCenterFromBackground_AnAzimuthalShadowIsNotACentreError", "[BeamC
     CHECK(estimate->beam_x_pxl == Catch::Approx(geom_true.GetBeamX_pxl()).margin(1.0));
     CHECK(estimate->beam_y_pxl == Catch::Approx(geom_true.GetBeamY_pxl()).margin(1.0));
 }
+
+// The same, with the detector tilted. Every pixel's 2-theta and azimuth, and the derivative of
+// 2-theta with respect to the centre that the fit is built on, go through the detector rotation
+// matrix, so the tilt is not a detail of the geometry here - it is in the Jacobian.
+TEST_CASE("BeamCenterFromBackground_SurvivesADetectorTilt", "[BeamCenter]") {
+    DiffractionExperiment x = TestExperiment();
+    x.PoniRot1_rad(0.005f).PoniRot2_rad(-0.003f);
+    PixelMask pixel_mask(x);
+
+    const DiffractionGeometry geom_true = OffsetBy(x.GetDiffractionGeometry(), 3.0f, -2.5f);
+    const auto [direct_x, direct_y] = geom_true.GetDirectBeam_pxl();
+    REQUIRE(std::hypot(direct_x - geom_true.GetBeamX_pxl(), direct_y - geom_true.GetBeamY_pxl()) > 5.0f);
+
+    const auto projection = SynthesiseProjection(x, pixel_mask, geom_true, 60.0f, 1.0f);
+    const auto estimate = FindBeamCenterFromBackground(x, pixel_mask, projection);
+
+    REQUIRE(estimate.has_value());
+    CHECK(estimate->beam_x_pxl == Catch::Approx(geom_true.GetBeamX_pxl()).margin(0.5));
+    CHECK(estimate->beam_y_pxl == Catch::Approx(geom_true.GetBeamY_pxl()).margin(0.5));
+    CHECK(estimate->sigma_pxl < 1.0f);
+}
diff --git a/tests/BeamCenterFromSpotsTest.cpp b/tests/BeamCenterFromSpotsTest.cpp
index e5ee73d99..fa4102621 100644
--- a/tests/BeamCenterFromSpotsTest.cpp
+++ b/tests/BeamCenterFromSpotsTest.cpp
@@ -282,3 +282,32 @@ TEST_CASE("BeamCenterFromSpots_SurvivesASpindleOffPerpendicular", "[BeamCenter]"
     CHECK(std::hypot(uncorrected->beam_x_pxl - truth.GetBeamX_pxl(),
                      uncorrected->beam_y_pxl - truth.GetBeamY_pxl()) > 3.0f * corrected_error);
 }
+
+// No detector is mounted exactly square to the beam, and a tilt separates the two centres this
+// estimator works with: the PONI it reports and the DIRECT BEAM both mirror lines are taken about.
+// The conversion between them is exact - it is where the vote is centred, where each tooth's
+// refinement starts, and how the answer is turned back out of the spindle frame - so a sign error
+// in it is a sign error of twice the offset in the answer. Here that offset is about 12 px, which
+// is 24 times the tolerance below.
+TEST_CASE("BeamCenterFromSpots_SurvivesADetectorTilt", "[BeamCenter]") {
+    DiffractionExperiment x = TestExperiment(Coord(1, 0, 0));
+    x.PoniRot1_rad(0.005f).PoniRot2_rad(-0.003f);
+    const DiffractionGeometry truth = OffsetBy(x.GetDiffractionGeometry(), 3.0f, -2.5f);
+    const Sweep sweep = MakeSweep(x, truth, PAIRS);
+
+    // The tilt has to be worth testing: with the two centres on top of each other the conversion
+    // is the identity and its sign is unobservable.
+    const auto [direct_x, direct_y] = truth.GetDirectBeam_pxl();
+    REQUIRE(std::hypot(direct_x - truth.GetBeamX_pxl(), direct_y - truth.GetBeamY_pxl()) > 5.0f);
+
+    SpindleEstimate fitted;
+    const auto estimate = FindBeamCenterFromSpotSymmetry(x, sweep.angle_deg, sweep.spots, &fitted);
+
+    REQUIRE(estimate.has_value());
+    CHECK(estimate->beam_x_pxl == Catch::Approx(truth.GetBeamX_pxl()).margin(0.5));
+    CHECK(estimate->beam_y_pxl == Catch::Approx(truth.GetBeamY_pxl()).margin(0.5));
+    CHECK(estimate->sigma_pxl < 1.0f);
+    // The spindle is perpendicular to the beam and on a lab axis here, so the tilt must not be
+    // read as an azimuth: what the fit sees of the detector belongs to the detector.
+    CHECK(fitted.azimuth_rad == Catch::Approx(0.0f).margin(5e-4));
+}