diff --git a/contrib/test_decoding/expected/rewrite.out b/contrib/test_decoding/expected/rewrite.out
index 3bf2afa931..28998b86f9 100644
--- a/contrib/test_decoding/expected/rewrite.out
+++ b/contrib/test_decoding/expected/rewrite.out
@@ -103,6 +103,10 @@ COMMIT;
  -- repeated rewrites in different transactions
 VACUUM FULL pg_class;
 VACUUM FULL pg_class;
+-- reindexing of important relations / indexes
+REINDEX TABLE pg_class;
+REINDEX INDEX pg_class_oid_index;
+REINDEX INDEX pg_class_tblspc_relfilenode_index;
 INSERT INTO replication_example(somedata, testcolumn1) VALUES (5, 3);
 BEGIN;
 INSERT INTO replication_example(somedata, testcolumn1) VALUES (6, 4);
diff --git a/contrib/test_decoding/sql/rewrite.sql b/contrib/test_decoding/sql/rewrite.sql
index 4271b82bea..c9503a0da5 100644
--- a/contrib/test_decoding/sql/rewrite.sql
+++ b/contrib/test_decoding/sql/rewrite.sql
@@ -74,6 +74,11 @@ COMMIT;
 VACUUM FULL pg_class;
 VACUUM FULL pg_class;
 
+-- reindexing of important relations / indexes
+REINDEX TABLE pg_class;
+REINDEX INDEX pg_class_oid_index;
+REINDEX INDEX pg_class_tblspc_relfilenode_index;
+
 INSERT INTO replication_example(somedata, testcolumn1) VALUES (5, 3);
 
 BEGIN;
diff --git a/src/backend/catalog/index.c b/src/backend/catalog/index.c
index 331f90528c..ce024101fc 100644
--- a/src/backend/catalog/index.c
+++ b/src/backend/catalog/index.c
@@ -3316,28 +3316,34 @@ reindex_index(Oid indexId, bool skip_constraint_checks, char persistence,
 	 */
 	TransferPredicateLocksToHeapRelation(iRel);
 
+	/* Fetch info needed for index_build */
+	indexInfo = BuildIndexInfo(iRel);
+
+	/* If requested, skip checking uniqueness/exclusion constraints */
+	if (skip_constraint_checks)
+	{
+		if (indexInfo->ii_Unique || indexInfo->ii_ExclusionOps != NULL)
+			skipped_constraint = true;
+		indexInfo->ii_Unique = false;
+		indexInfo->ii_ExclusionOps = NULL;
+		indexInfo->ii_ExclusionProcs = NULL;
+		indexInfo->ii_ExclusionStrats = NULL;
+	}
+
+	/*
+	 * Build a new physical relation for the index. Need to do that before
+	 * "officially" starting the reindexing with SetReindexProcessing -
+	 * otherwise the necessary pg_class changes cannot be made with
+	 * encountering assertions.
+	 */
+	RelationSetNewRelfilenode(iRel, persistence);
+
+	/* ensure SetReindexProcessing state isn't leaked */
 	PG_TRY();
 	{
 		/* Suppress use of the target index while rebuilding it */
 		SetReindexProcessing(heapId, indexId);
 
-		/* Fetch info needed for index_build */
-		indexInfo = BuildIndexInfo(iRel);
-
-		/* If requested, skip checking uniqueness/exclusion constraints */
-		if (skip_constraint_checks)
-		{
-			if (indexInfo->ii_Unique || indexInfo->ii_ExclusionOps != NULL)
-				skipped_constraint = true;
-			indexInfo->ii_Unique = false;
-			indexInfo->ii_ExclusionOps = NULL;
-			indexInfo->ii_ExclusionProcs = NULL;
-			indexInfo->ii_ExclusionStrats = NULL;
-		}
-
-		/* We'll build a new physical relation for the index */
-		RelationSetNewRelfilenode(iRel, persistence);
-
 		/* Initialize the index and rebuild */
 		/* Note: we do not need to re-establish pkey setting */
 		index_build(heapRelation, iRel, indexInfo, true, true);
diff --git a/src/backend/catalog/indexing.c b/src/backend/catalog/indexing.c
index 0c994122d8..507879ecaf 100644
--- a/src/backend/catalog/indexing.c
+++ b/src/backend/catalog/indexing.c
@@ -82,9 +82,15 @@ CatalogIndexInsert(CatalogIndexState indstate, HeapTuple heapTuple)
 	Datum		values[INDEX_MAX_KEYS];
 	bool		isnull[INDEX_MAX_KEYS];
 
-	/* HOT update does not require index inserts */
+	/*
+	 * HOT update does not require index inserts. But with asserts enabled we
+	 * want to check that it'd be legal to currently insert into the
+	 * table/index.
+	 */
+#ifndef USE_ASSERT_CHECKING
 	if (HeapTupleIsHeapOnly(heapTuple))
 		return;
+#endif
 
 	/*
 	 * Get information from the state structure.  Fall out if nothing to do.
@@ -107,8 +113,10 @@ CatalogIndexInsert(CatalogIndexState indstate, HeapTuple heapTuple)
 	for (i = 0; i < numIndexes; i++)
 	{
 		IndexInfo  *indexInfo;
+		Relation	index;
 
 		indexInfo = indexInfoArray[i];
+		index = relationDescs[i];
 
 		/* If the index is marked as read-only, ignore it */
 		if (!indexInfo->ii_ReadyForInserts)
@@ -121,9 +129,18 @@ CatalogIndexInsert(CatalogIndexState indstate, HeapTuple heapTuple)
 		Assert(indexInfo->ii_Expressions == NIL);
 		Assert(indexInfo->ii_Predicate == NIL);
 		Assert(indexInfo->ii_ExclusionOps == NULL);
-		Assert(relationDescs[i]->rd_index->indimmediate);
+		Assert(index->rd_index->indimmediate);
 		Assert(indexInfo->ii_NumIndexKeyAttrs != 0);
 
+		/* see earlier check above */
+#ifdef USE_ASSERT_CHECKING
+		if (HeapTupleIsHeapOnly(heapTuple))
+		{
+			Assert(!ReindexIsProcessingIndex(RelationGetRelid(index)));
+			continue;
+		}
+#endif /* USE_ASSERT_CHECKING */
+
 		/*
 		 * FormIndexDatum fills in its values and isnull parameters with the
 		 * appropriate values for the column(s) of the index.
diff --git a/src/test/regress/expected/create_index.out b/src/test/regress/expected/create_index.out
index 326dc44177..ef8c8cd654 100644
--- a/src/test/regress/expected/create_index.out
+++ b/src/test/regress/expected/create_index.out
@@ -1939,6 +1939,24 @@ INFO:  index "reindex_verbose_pkey" was reindexed
 \set VERBOSITY default
 DROP TABLE reindex_verbose;
 --
+-- check that system tables can be reindexed
+--
+-- whole tables
+REINDEX TABLE pg_class; -- mapped, non-shared, critical
+REINDEX TABLE pg_index; -- non-mapped, non-shared, critical
+REINDEX TABLE pg_operator; -- non-mapped, non-shared, critical
+REINDEX TABLE pg_database; -- mapped, shared, critical
+REINDEX TABLE pg_shdescription; -- mapped, shared non-critical
+-- Check that individual system indexes can be reindexed. That's a bit
+-- different from the entire-table case because reindex_relation
+-- treats e.g. pg_class special.
+REINDEX INDEX pg_class_oid_index; -- mapped, non-shared, critical
+REINDEX INDEX pg_class_relname_nsp_index; -- mapped, non-shared, non-critical
+REINDEX INDEX pg_index_indexrelid_index; -- non-mapped, non-shared, critical
+REINDEX INDEX pg_index_indrelid_index; -- non-mapped, non-shared, non-critical
+REINDEX INDEX pg_database_oid_index; -- mapped, shared, critical
+REINDEX INDEX pg_shdescription_o_c_index; -- mapped, shared, non-critical
+--
 -- REINDEX CONCURRENTLY
 --
 CREATE TABLE concur_reindex_tab (c1 int);
diff --git a/src/test/regress/sql/create_index.sql b/src/test/regress/sql/create_index.sql
index f29b8ca826..f6f2d7c14b 100644
--- a/src/test/regress/sql/create_index.sql
+++ b/src/test/regress/sql/create_index.sql
@@ -747,6 +747,27 @@ REINDEX (VERBOSE) TABLE reindex_verbose;
 \set VERBOSITY default
 DROP TABLE reindex_verbose;
 
+--
+-- check that system tables can be reindexed
+--
+
+-- whole tables
+REINDEX TABLE pg_class; -- mapped, non-shared, critical
+REINDEX TABLE pg_index; -- non-mapped, non-shared, critical
+REINDEX TABLE pg_operator; -- non-mapped, non-shared, critical
+REINDEX TABLE pg_database; -- mapped, shared, critical
+REINDEX TABLE pg_shdescription; -- mapped, shared non-critical
+
+-- Check that individual system indexes can be reindexed. That's a bit
+-- different from the entire-table case because reindex_relation
+-- treats e.g. pg_class special.
+REINDEX INDEX pg_class_oid_index; -- mapped, non-shared, critical
+REINDEX INDEX pg_class_relname_nsp_index; -- mapped, non-shared, non-critical
+REINDEX INDEX pg_index_indexrelid_index; -- non-mapped, non-shared, critical
+REINDEX INDEX pg_index_indrelid_index; -- non-mapped, non-shared, non-critical
+REINDEX INDEX pg_database_oid_index; -- mapped, shared, critical
+REINDEX INDEX pg_shdescription_o_c_index; -- mapped, shared, non-critical
+
 --
 -- REINDEX CONCURRENTLY
 --