diff --git a/src/join.jl b/src/join.jl
index 37ee57c..c809fde 100644
--- a/src/join.jl
+++ b/src/join.jl
@@ -33,11 +33,15 @@ function join(
     error("Not all DataFrames have all columns given by `cols`")
   end
 
+  # Work with local copies to avoid mutating caller-provided arrays
+  cols = copy(cols)
+  invariant_cols = copy(invariant_cols)
+
   if :id in cols
     deleteat!(cols, findall(cols .== :id))
   end
   if :id in invariant_cols
-    deleteat!(cols, findall(cols .== :id))
+    deleteat!(invariant_cols, findall(invariant_cols .== :id))
   end
   invariant_cols = [:id; invariant_cols]
   cols = setdiff(cols, invariant_cols)
@@ -48,6 +52,10 @@ function join(
 
   s = first(stats)[1]
   df = stats[s][:, invariant_cols]
+  # Apply hdr_override to invariant columns, but keep :id stable.
+  inv_col_names = [c == :id ? "id" : (haskey(hdr_override, c) ? hdr_override[c] : String(c)) for c in invariant_cols]
+  rename!(df, Dict(c => Symbol(n) for (c, n) in zip(invariant_cols, inv_col_names)))
+  invariant_cols = [Symbol(n) for n in inv_col_names]
 
   rename_f(c, s) = begin
     symbol_c = Symbol(c)
diff --git a/test/test-tables.jl b/test/test-tables.jl
index c0dd974..bafb00d 100644
--- a/test/test-tables.jl
+++ b/test/test-tables.jl
@@ -42,12 +42,33 @@ cols = [:status, :name, :f, :t, :iter]
       stats,
       [:status, :f, :t],
       invariant_cols = [:name],
-      hdr_override = Dict(:status => "flag"),
+      hdr_override = Dict(:status => "flag", :name => "Name"),
     )
 
+    @test :Name in propertynames(df_joined)
+    @test :name ∉ propertynames(df_joined)
     println(df_joined)
   end
 
+  @testset "join ignores id header override" begin
+    df_joined = join(
+      stats,
+      [:status, :f, :t],
+      invariant_cols = [:name],
+      hdr_override = Dict(:id => "identifier", :status => "flag"),
+    )
+    df_expected = join(
+      stats,
+      [:status, :f, :t],
+      invariant_cols = [:name],
+      hdr_override = Dict(:status => "flag"),
+    )
+
+    @test :id in propertynames(df_joined)
+    @test :identifier ∉ propertynames(df_joined)
+    @test df_joined == df_expected
+  end
+
   @testset "joined results in latex format" begin
     df_joined = join(
       stats,