What is the correct view regarding rinsing the mouth (madhamadha), does it invalid the fast?
No, it (rinsing the mouth) does not break the fast. It has only been reported regarding rinsing the nose (istinshāq).
Istinshāq is legalisated for the fasting person, except he is prohibited from what? – from exaggerating.
“بَالِغْ فِي الِاسْتِنْشَاقِ، إِلَّا أَنْ تَكُونَ صَائِمًا”
Exaggerate in istinshāq except if you are fasting. 
It is permissible to rinse the mouth and it is permissible to rinse the nose (without exaggeration).
Footnote:  Narrated by Āsim Ibn Laqēt Ibn Sabera in Abū Dāwūd (2366)
Answered by: Shaikh Abu Abdillah Muhammad Ba Jamaal (May Allah preserve him)
Translated by: Abu Sufyān Sāmi ibn Daniel Al-Ghāni