diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index 531d8d7a3917e..46e6486d37c20 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -7,6 +7,7 @@
                                       ensure_index_from_sequences,
                                       InvalidIndexError)  # noqa
 from pandas.core.indexes.category import CategoricalIndex  # noqa
+from pandas.core.indexes.frozen import FrozenList
 from pandas.core.indexes.multi import MultiIndex  # noqa
 from pandas.core.indexes.interval import IntervalIndex  # noqa
 from pandas.core.indexes.numeric import (NumericIndex, Float64Index,  # noqa
@@ -95,9 +96,7 @@ def _get_combined_index(indexes, intersect=False, sort=False):
     elif len(indexes) == 1:
         index = indexes[0]
     elif intersect:
-        index = indexes[0]
-        for other in indexes[1:]:
-            index = index.intersection(other)
+        index = _intersect_indexes(indexes)
     else:
         index = _union_indexes(indexes, sort=sort)
         index = ensure_index(index)
@@ -110,6 +109,17 @@ def _get_combined_index(indexes, intersect=False, sort=False):
     return index
 
 
+def _intersect_indexes(indexes):
+    """Return the intersection of indexes
+
+    Preserves the order of the first index.
+    """
+    index = indexes[0]
+    for other in indexes[1:]:
+        index = index.intersection(other)
+    return index
+
+
 def _union_indexes(indexes, sort=True):
     """Return the union of indexes
 
@@ -125,15 +135,6 @@ def _union_indexes(indexes, sort=True):
     -------
     Index
     """
-    if len(indexes) == 0:
-        raise AssertionError('Must have at least 1 Index to union')
-    if len(indexes) == 1:
-        result = indexes[0]
-        if isinstance(result, list):
-            result = Index(sorted(result))
-        return result
-
-    indexes, kind = _sanitize_and_check(indexes)
 
     def _unique_indices(inds):
         """Convert indexes to lists and concatenate them, removing duplicates
@@ -153,35 +154,46 @@ def conv(i):
                 i = i.tolist()
             return i
 
-        return Index(
-            lib.fast_unique_multiple_list([conv(i) for i in inds], sort=sort))
+        inds = [conv(i) for i in inds]
+        ind_values = lib.fast_unique_multiple_list(inds, sort=sort)
+        return Index(ind_values)
 
-    if kind == 'special':
+    if len(indexes) == 0:
+        raise AssertionError('Must have at least 1 Index to union')
+    if len(indexes) == 1:
         result = indexes[0]
+        if isinstance(result, list):
+            result = Index(sorted(result))
+        return result
+
+    indexes, kind = _sanitize_and_check(indexes)
 
+    if kind == 'special':
+
+        result = indexes[0]
         if hasattr(result, 'union_many'):
             return result.union_many(indexes[1:])
         else:
             for other in indexes[1:]:
                 result = result.union(other)
             return result
-    elif kind == 'array':
-        index = indexes[0]
-        for other in indexes[1:]:
-            if not index.equals(other):
 
-                if sort is None:
-                    # TODO: remove once pd.concat sort default changes
-                    warnings.warn(_sort_msg, FutureWarning, stacklevel=8)
-                    sort = True
+    elif kind == 'array':
 
-                return _unique_indices(indexes)
+        if _all_indexes_same(indexes):
+            index = indexes[0]
+            name = _get_consensus_names(indexes)[0]
+            if name != index.name:
+                index = index._shallow_copy(name=name)
+            return index
+        else:
+            if sort is None:
+                # TODO: remove once pd.concat sort default changes
+                warnings.warn(_sort_msg, FutureWarning, stacklevel=8)
+                sort = True
+            return _unique_indices(indexes)
 
-        name = _get_consensus_names(indexes)[0]
-        if name != index.name:
-            index = index._shallow_copy(name=name)
-        return index
-    else:  # kind='list'
+    else:  # kind == 'list'
         return _unique_indices(indexes)
 
 
@@ -205,18 +217,18 @@ def _sanitize_and_check(indexes):
     sanitized_indexes: list of Index or array-like objects
     type: {'list', 'array', 'special'}
     """
-    kinds = list({type(index) for index in indexes})
+    kinds = {type(index) for index in indexes}
 
     if list in kinds:
-        if len(kinds) > 1:
-            indexes = [Index(com.try_sort(x))
-                       if not isinstance(x, Index) else
-                       x for x in indexes]
-            kinds.remove(list)
-        else:
+        if len(kinds) == 1:
             return indexes, 'list'
+        else:
+            indexes = [Index(com.try_sort(i))
+                       if not isinstance(i, Index) else i
+                       for i in indexes]
+            kinds.remove(list)
 
-    if len(kinds) > 1 or Index not in kinds:
+    if any(kind != Index for kind in kinds):
         return indexes, 'special'
     else:
         return indexes, 'array'
@@ -237,14 +249,11 @@ def _get_consensus_names(indexes):
     list
         A list representing the consensus 'names' found
     """
-
-    # find the non-none names, need to tupleify to make
-    # the set hashable, then reverse on return
-    consensus_names = {tuple(i.names) for i in indexes
-                       if com._any_not_none(*i.names)}
-    if len(consensus_names) == 1:
-        return list(list(consensus_names)[0])
-    return [None] * indexes[0].nlevels
+    non_empty = {i.names for i in indexes if com._any_not_none(*i.names)}
+    if len(non_empty) == 1:
+        return non_empty.pop()
+    else:
+        return FrozenList([None] * indexes[0].nlevels)
 
 
 def _all_indexes_same(indexes):
@@ -260,7 +269,5 @@ def _all_indexes_same(indexes):
         True if all indexes contain the same elements, False otherwise
     """
     first = indexes[0]
-    for index in indexes[1:]:
-        if not first.equals(index):
-            return False
-    return True
+    others = indexes[1:]
+    return all(first.equals(other) for other in others)