Praise be to Allah.
Turning to the right and left when saying Hayya ‘ala as-salaah, hayya ‘ala al-falaah in the adhaan is Sunnah, because of the report narrated by Muslim (777) from Abu Juhayfah, who said: I came to the Prophet (blessings and peace of Allah be upon him) in Makkah when he was in al-Abtah, in a red leather tent of his. Bilaal brought his water for wudoo’, and the Prophet (blessings and peace of Allah be upon him) came out, wearing a red hullah (suit). It is as if I can see the whiteness of his shins. Then he did wudoo’, and people rushed to catch his water for barakah (blessing), to sprinkle it on themselves or others. Then Bilaal gave the call to prayer, and I watched his mouth moving to this side and that, to the right and the left as he said, Hayya ‘ala as-salaah, hayya ‘ala al-falaah.
A shorter version was narrated by al-Bukhaari (598).
According to an-Nasaa’i: Bilaal came out and gave the call to prayer, and did like this in his adhaan – turning to the right and to the left.
This turning (to the right and to the left) is so as to make the voice carry further, because the mu’adhdhin used to give the adhaan from a high place, so that his voice would carry further.
Hence some other scholars were of the view that if he gives the adhaan into a microphone or loudspeaker, he does not have to turn to the right and the left.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
We should point out that the wisdom behind turning to the right and to the left is so that the call will reach those to whom it is addressed on the right and on the left. Based on that, the one who gives the adhaan into a microphone or loudspeaker should not turn to the right and to the left, because conveying of the sound will be done via the loudspeakers on the minaret, and if he were to turn, the sound would become more faint, because he would be moving away from the microphone.
End quote from ash-Sharh al-Mumti‘ (2/60).
And he (may Allah have mercy on him) said:
What I think with regard to loudspeakers that are used nowadays is that he (the mu’adhdhin) should not turn to the right or to the left, either when saying Hayya ‘ala as-salaah or when saying Hayya ‘ala al-falaah. Nowadays instead of turning, the loudspeakers should be placed appropriately, on the minaret, one on the right and one on the left.
End quote from al-Liqa’ al-Maftooh (155/17).
Shaykh Ibn Jibreen (may Allah have mercy on him) said:
It is Sunnah for the mu’adhdhin to turn to the right and to the left when saying Hayya ‘ala as-salaah, hayya ‘ala al-falaah, so that those who are on those two sides and behind him can hear. But perhaps this only applies when the adhaan is given from the top of the minaret and there is no loudspeaker, as was usually the case in the past. But if the adhaan is given into the microphone, I think that there is no need to turn to the right and to the left, because the microphone is in front of the mu’adhdhin. So if he faces it, his voice will be strong, because if he turns away from it his voice will become more faint, and the mu’adhdhin is instructed to make his voice loud and strong.
End quote from Fataawa Islamiyyah
Shaykh ‘Abd ar-Razzaaq ‘Afeefi (may Allah have mercy on him) was asked about giving the adhaan outside the mosque.
What matters is making it heard, so whatever is more effective in conveying the sound is more appropriate. If they are equally effective, then there is no advantage to giving the adhaan outside the mosque. The same applies to one who turns to the right and to the left when saying Hayya ‘ala as-salaah, hayya ‘ala al-falaah, and his voice becomes more faint because he moves away from the mic, whereas the loudspeakers on the minaret broadcast the sound right and left. If the reason for doing a thing is no longer applicable, then it is no longer prescribed, as may be understood from the story of the difference of opinion among the Sahaabah (may Allah be pleased with them) with regard to ‘Asr prayer during the campaign to Banu Qurayzah.
End quote from Fataawa ash-Shaykh ‘Abd ar-Razzaaq ‘Afeefi (p. 16)
And Allah knows best.