Withdrawing my suggestion.

@@ -7,4 +6,0 @@
-</k_ele>
-<k_ele>
-<keb>半面</keb>
-<ke_inf>&sK;</ke_inf>
@@ -28 +23,0 @@
-<s_inf>occ. 半面</s_inf>

Reopen..

Adding my suggestion about 半面. I'll approve and reopen.

@@ -6,0 +7,4 @@
+</k_ele>
+<k_ele>
+<keb>半面</keb>
+<ke_inf>&sK;</ke_inf>
@@ -23,0 +28 @@
+<s_inf>occ. 半面</s_inf>

（１）Sankoku:
一｟名｠
①反対の（方）面。
「やさしさの━にたくましさを秘(ひ)めている」
②〔接続助詞的に用いて〕…のと同時に、別の面では。
「期待する━（で）、不安もある」
二｟接｠
別の面では。反対の面からみると。
「結果には満足している。━、反省もある」



（２）BCCWJ:
反面 has 1069 counts in total

For case 3:
｛〘活用形＝連体形〙｝｛反面｝＝ 456
｛〘活用形＝連体形〙｝｛反面｝｛、｝＝ 314

For case 2:
｛、｝｛反面｝＝ 117
｛、｝｛反面｝｛、｝＝ 68
｛。｝｛反面｝＝ 100
｛。｝｛反面｝｛、｝＝ 62
Note that there may be a few missing for the counts with commas, as some of the sentences used , instead of 、.

I scrolled through about half of the 1000 hits and didn't really spot usage for case 1. This doesn't mean it doesn't exist, but it's certainly far less common than case 2 and 3.

Other notable counts:
｛〘品詞＝連体詞〙｝｛反面｝＝ 212
｛反面｝｛教師｝＝ 64

Unidic generally classes 反面 as a 名詞-普通名詞-副詞可能



（３）Representations of the conjunctive use in other references:

Daijs: 2（副詞的に用いて）他の面から見る場合。他面。「水に強い―、熱に弱い」
Smk:（副）
その物事に関して、まず第一にあげられる面に対して、それとは対照的な面もありうると判断する様子。
「娘の結婚はうれしい━寂しくもある」
Koj: ②（副詞的に）反対の面から見て。「陽気な―、涙もろいところがある」

Though I believe they aren't quite sufficient.

.

Proposing a reordering of the senses, and an additional sense.

＞ Should we re-order the senses?
I think they should very much be reordered, but I'm not sure about that [rare] tag. 
In Anonさん's example 反面 is certainly a noun, forming a larger contrastive adverbial frame with ある面では.
ーーーーーーーーーーーーーーーーーーーーーーー

I believe there are three cases to distinguish here, if we want to be precise:
Case 1: the plain noun 反面 (which does appear to be rather uncommon)
Case 2: grammaticalized 反面 as a plain conjunction/conjunctive adverb, i.e. a *自立語*
Case 3: grammaticalized 反面 as a conjunctive particle taking a 連体修飾語, i.e. a *付属語*

This paradigm is presented in（１）
Counts for the separate cases are presented in（２）



Furthermore I would assert that in case 3, 反面 is most naturally translated as "while", "though", and so forth, but not really as "on the other hand".
While in case 2 (this includes その反面), it is something like "on the other hand" or "then again"
Take this sentence from Brian's quoted site as an example (https://nihongo-online.jp/net/index.php?key=bbjgig41h-81):

「大学が決まって、新しい生活に期待する『反面』、不安も大きい。」
"I’ve decided on my university, and while I'm anticipating my new life, I also have big worries."

This is also substantiated by DOJG examples + translations
ーーーーーーーーーーーーーーーーーーーーーーー

While looking at the corpus data I noticed that case 2 uses of 反面 are often preceded by another conjunction or conjunctive particle.
If there is interest, I could also provide counts for various of those cases.

ーーーーーーーーーーーーーーーーーーーーーーー

I've purposefully left out the 半面・反面 discussion here so as to not make it more confusing than it already is.
That should probably be the next step after this entry alone has been settled.

@@ -13,4 +13,7 @@
-<pos>&n;</pos>
-<misc>&rare;</misc>
-<gloss>opposite side</gloss>
-<gloss>reverse</gloss>
+<pos>&conj;</pos>
+<pos>&prt;</pos>
+<s_inf>after the attributive form of a verb or adjective, or the particle の</s_inf>
+<gloss>while</gloss>
+<gloss>although</gloss>
+<gloss>though</gloss>
+<gloss>but</gloss>
@@ -22,3 +25,10 @@
-<gloss>though</gloss>
-<gloss>although</gloss>
-<gloss>while</gloss>
+<gloss>at the same time</gloss>
+<gloss>then again</gloss>
+<gloss>and yet</gloss>
+</sense>
+<sense>
+<pos>&n;</pos>
+<misc>&rare;</misc>
+<gloss>opposite side</gloss>
+<gloss>other side</gloss>
+<gloss>reverse</gloss>

女の子ってのは、ある面では現実的で打算的だけど、反面では理想とか夢を持ち続けてるものなんだから。

yet, it really didn't take me a whole lot of time to find a sample for sense 1