Q:

Is there a name for kanji with the same ON and KUN readings?

Is there a name for kanji with the same on and kun reading? I mean kanji like 労.

A:

A few kanji do have the same on- and kun-readings. In the joyo-kanji list,

死

on: シ as in 死亡【シボウ】 (death), 死体【シタイ】 (corpse)
kun: 死【し】ぬ (to die)

架

on: カ as in 高架【コウカ】 (elevated/raised (road)), 担架【タンカ】 (stretcher)
kun: 架【か】ける (to build (a bridge))

I don't know the name for this type of kanji. These are probably mere coincidences and I wonder if this will help people learn Japanese.

A:

I'm not sure there are kanji that have the same on and kun reading and if there are, they would be extremely rare. For this to happen you would have to have a native Japanese word (at most two morae), which happens to be pronounced like a on reading (quite unlikely) and that word happens to be the on reading of the kanji that matches the meaning of the original Japanese word (extremely unlikely).
For example, か "mosquito" could be an on reading of a kanji. But it's not the on reading of the kanji that matches the meaning "mosquito", which would be 蚊【ブン】. So, we get 蚊 with kun reading か and on reading ブン.

It may happen that the on reading of a single kanji may be used as a word by itself (e.g. 駅). That doesn't make the reading a kun reading.
Also, sometimes the on reading of a kanji may be used in verbs (e.g. 愛す or 転じる). Even if the verb can be written in hiragana (as in あいす or てんじる), that doesn't make the reading (あい or てん) a kun reading. These verbs, that look like

[on reading] + す
  [on reading] + じる
  [on reading] + ずる

are fossilized versions of

[on reading] + する

as in 勉強する "to study" and you will agree that べんきょう isn't a kun reading of 勉強 just because it appears in べんきょうする.

