The story of how the Prophet (ﷺ) inverted his cloak is mentioned in Sahih al-Bukhari from the narration of 'Abdullah bin Zaid. And it contains (the words):

"He (ﷺ) faced the Qiblah making supplication. Then, he prayed two Rak'at, reciting (the Qur'an) in them audibly."

وَقِصَّةُ اَلتَّحْوِيلِ فِي

اَلصَّحِيحِ

مِنْ: حَدِيثِ عَبْدِ اَللَّهِ بْنِ زَيْدٍ، وَفِيهِ: { فَتَوَجَّهَ إِلَى اَلْقِبْلَةِ, يَدْعُو, ثُمَّ صَلَّى رَكْعَتَيْنِ, جَهَرَ فِيهِمَا بِالْقِرَاءَةِ } [1]‏ .‏