diff --git a/hll/include/Hll8Array-internal.hpp b/hll/include/Hll8Array-internal.hpp
index 2c5823a6..14a2bbd7 100644
--- a/hll/include/Hll8Array-internal.hpp
+++ b/hll/include/Hll8Array-internal.hpp
@@ -98,6 +98,9 @@ void Hll8Array<A>::internalCouponUpdate(uint32_t coupon) {
 
   const uint8_t curVal = this->hllByteArr_[slotNo];
   if (newVal > curVal) {
+    // A prior HLL merge may have left the estimator state stale. Rebuild it before applying
+    // an incremental update for a changed register.
+    this->check_rebuild_kxq_cur_min();
     this->hllByteArr_[slotNo] = newVal;
     this->hipAndKxQIncrementalUpdate(curVal, newVal);
     this->numAtCurMin_ -= curVal == 0; // interpret numAtCurMin as num zeros
diff --git a/hll/include/HllArray-internal.hpp b/hll/include/HllArray-internal.hpp
index 62ea7f78..92dbe8b2 100644
--- a/hll/include/HllArray-internal.hpp
+++ b/hll/include/HllArray-internal.hpp
@@ -465,6 +465,10 @@ bool HllArray<A>::isCompact() const {
 
 template<typename A>
 bool HllArray<A>::isEmpty() const {
+  // mergeHll() is only called with non-empty sketches and sets this flag after updating the
+  // register array. The cached curMin_/numAtCurMin_ may still have their empty-sketch values,
+  // but a pending rebuild therefore proves that this array is not empty.
+  if (rebuild_kxq_curmin_) return false;
   const uint32_t configK = 1 << this->lgConfigK_;
   return (curMin_ == 0) && (numAtCurMin_ == configK);
 }
diff --git a/hll/include/HllUnion-internal.hpp b/hll/include/HllUnion-internal.hpp
index 27adab74..a469a2d0 100644
--- a/hll/include/HllUnion-internal.hpp
+++ b/hll/include/HllUnion-internal.hpp
@@ -39,6 +39,9 @@ hll_union_alloc<A>::hll_union_alloc(uint8_t lg_max_k, const A& allocator):
 
 template<typename A>
 hll_sketch_alloc<A> hll_union_alloc<A>::get_result(target_hll_type target_type) const {
+  if (gadget_.sketch_impl->getCurMode() == hll_mode::HLL) {
+    static_cast<HllArray<A>*>(gadget_.sketch_impl)->check_rebuild_kxq_cur_min();
+  }
   return hll_sketch_alloc<A>(gadget_, target_type);
 }
 
diff --git a/hll/test/HllUnionTest.cpp b/hll/test/HllUnionTest.cpp
index ceaef12f..1d065d19 100644
--- a/hll/test/HllUnionTest.cpp
+++ b/hll/test/HllUnionTest.cpp
@@ -314,4 +314,74 @@ TEST_CASE("hll union: check hll to hll", "[hll_union]") {
   union_two_sketches_with_overlap(1000000, 11, HLL_4);
 }
 
+static hll_sketch make_hll_sketch(uint8_t lg_config_k, uint64_t start, uint64_t end) {
+  hll_sketch sketch(lg_config_k);
+  for (uint64_t i = start; i < end; ++i) sketch.update(i);
+  return sketch;
+}
+
+static hll_sketch::vector_bytes serialize_flat_union(
+    const hll_sketch& first, const hll_sketch& second, const hll_sketch& third) {
+  hll_union u(8);
+  u.update(first);
+  u.update(second);
+  u.update(third);
+  return u.get_result(HLL_8).serialize_updatable();
+}
+
+static hll_sketch::vector_bytes serialize_nested_union(
+    const hll_sketch& first, const hll_sketch& second, const hll_sketch& third) {
+  hll_union prefix(8);
+  prefix.update(first);
+  prefix.update(second);
+
+  hll_union u(8);
+  u.update(prefix.get_result(HLL_8));
+  u.update(third);
+  return u.get_result(HLL_8).serialize_updatable();
+}
+
+TEST_CASE("hll union: downsampling merge is not empty", "[hll_union]") {
+  const hll_sketch sketch = make_hll_sketch(15, 0, 100000);
+  hll_union u(8);
+  u.update(sketch);
+  REQUIRE_FALSE(u.is_empty());
+}
+
+TEST_CASE("hll union: mixed lgConfigK estimate is merge-order independent", "[hll_union]") {
+  const uint64_t n = 100000;
+  const hll_sketch a = make_hll_sketch(15, 0, n);
+  const hll_sketch b = make_hll_sketch(8, n, 2 * n);
+  const double truth = 2.0 * n;
+
+  hll_union larger_first(8);
+  larger_first.update(a);
+  larger_first.update(b);
+  REQUIRE(larger_first.get_estimate() == Approx(truth).epsilon(0.1));
+
+  hll_union smaller_first(8);
+  smaller_first.update(b);
+  smaller_first.update(a);
+  REQUIRE(smaller_first.get_estimate() == Approx(truth).epsilon(0.1));
+}
+
+TEST_CASE("hll union: scalar update after downsampling merge", "[hll_union]") {
+  const uint64_t n = 100000;
+  const hll_sketch sketch = make_hll_sketch(15, 0, n);
+  hll_union u(8);
+  u.update(sketch);
+  for (uint64_t i = n; i < 2 * n; ++i) u.update(i);
+  REQUIRE(u.get_estimate() == Approx(2.0 * n).epsilon(0.1));
+}
+
+TEST_CASE("hll union: serialization is grouping independent", "[hll_union]") {
+  const uint64_t n = 100000;
+  const hll_sketch a = make_hll_sketch(15, 0, n);
+  const hll_sketch b = make_hll_sketch(8, n, 2 * n);
+  const hll_sketch c = make_hll_sketch(11, n / 2, n + n / 2);
+  REQUIRE(serialize_nested_union(b, c, a) == serialize_flat_union(b, c, a));
+  REQUIRE(serialize_nested_union(a, c, b) == serialize_flat_union(a, c, b));
+  REQUIRE(serialize_nested_union(a, b, c) == serialize_flat_union(a, b, c));
+}
+
 } /* namespace datasketches */