Srila Prabupada mentions that either of Hare Krishna or Hare Rama can be chanted first.
"There is no difference between the holy name of Krishna and Rama. But still, there is some distinction. In the śāstra it is said that if you chant one thousand times Lord Viṣṇu's name, that is equal to chanting one name of Lord Rāma. Sahasra, Sahasra, Viṣṇu-Sahasra-Nāma. And if you chant three times Lord Rāma's name, it is equal to one time chanting Kṛṣṇa's name. This is the shastric injunction."
- Srila Prabhupada (Lecture on SB 1.2.5,
Aligarh, October 9, 1976)
In practice, however, the question of order of either ‘Hare Rama’ first or ‘Hare Krishna’ first, is not of primary importance, since as soon as one starts chanting the mantra, such differences disappear in the continuous flow of the names.
Valmiki Muni used to be a dreaded dacoit before meeting Narada Muni. Narada Muni gave him mantra ‘Ra-Ma’ but Valmiki Muni wasn’t able to chant it owing to lack of purity, hence Narada Muni inverted the mantra to ‘Ma-Ra Ma-Ra’ instead of ‘Ra-Ma’ so that Valmiki Muni can be delivered! When one chants in flow ‘Ma-Ra Ma-Ra Ma-Ra’ they are automatically chanting Ra-Ma! The meeting with a Rasika Vaishnav changes the course of one’s destiny, no matter their past! So, as mentioned earlier as well, as soon as one starts chanting the mantra, such differences disappear in the continuous flow of the names.
The Lord or His confidential associates are always trying to deliver the fallen souls from the repetition of birth and death.
Prior to the appearance of most merciful incarnation Lord Chaitanya Mahaprabhu in this age of Kali, there were Yugas like Satya Yuga, Treta Yuga and Dwapara Yuga in which people had some level of purity.
But in this age of Kali, people have short lives. They are confused, angry, fight like crazy, and are in mode of ignorance.
According to the Srimad Bhagavatam, it is said, kalau sudra sambhavaḥ. It means that in this age of Kali, everyone will be like shudras. They do not have any tinge of qualification to study the original Vedas. All of this means that there is hardly any room for their deliverance.
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Though, in this Kali Yuga, everyone, no matter even if they are shudras, they do have the right to study agamas and vaiṣhnava tantras. In prominent vaiṣṇava-agamas or satvata-tantras, it is said that all can chant Hare Krishna Mahamantra (beginning with ‘Hare Krishna’)!
This is sync with Srimad Bhagavatam (11.5.31) where it is well stated that in Kali Yuga, thevaishnava tantras are prominent, which exclusively tells to chant the holy names of the Lord in form of Hare Krishna Mahamantra beginning with ‘Hare Krishna’!
According to the Kali-Santaraṇa Upaniṣad, in Hare Krishna Mahamantra, Hare Krishna is stated first.
Kṛishnadasa Babaji (initial 20th century), a popular Rasika Vaishnav, writes in his book Sri Maha-Mantra-Vyakhyaṣṭakam that in Kali Yuga, there are no brahmans, and all are like shudras. As all are shudras, they are not qualified or eligible to read Vedas or Upanishad, and thus by default, they are not eligible to read Kali Santarana Upanishad as well. Thus there is no point in following mantras mentioned there as one is not eligible to read it only.
Then Krishnadasa Babaji writes that, athough, tantras and agamas which all can read mention about chanting Hare Krishna Mahamantra starting with ‘Hare Krishna’! So, every single person on earth can chant Hare Krishna Mahamantra, as mentioned in agamas and vaishnav tantras!
Sri Caitanya Mahaprabhu preached the Hare Krishna Mahamantra beginning with ‘Hare Krishna’ from the agamas, thus providing his mercy to everyone, and at the same time respecting the rules given in the Vedas regarding the eligibility.
However, in the editions of Kali-Santaraṇa Upaniṣad that is in circulation today, Hare Krishna Mahamantra has a reversed reading, with the ‘Hare Rama’ part coming first instead of ‘Hare Krishna’.
Speaking about the order of the Mahamantra in Kali-Santaraṇa Upaniṣad, Srila Bhaktisiddhanta Saraswati Thakur Prabhupada said:
“Kali-Santaraṇa Upaniṣad has been published in Mumbai and Madras, but because the manuscripts of the Upaniṣad were collected from the adulterated members of the Rāmāyet-sampradāya, the order of the Mahamantra in these particular editions was reversed. But even despite being so, its meaning and position cannot be reversed. Any wise person will not accept any order or reading different from the mahamantra and its order given by the person denoted by the holy name (nāmi), Śrī Gaurasundara, who himself descended in Kali-yuga to distribute the easy process for crossing over the age of Kali (Kali-Santaraṇa) and attaining love of God.”
According to Radha-Tantra, Brahma-Yamala, Ananta-Samhita, Paippalada-Sakha of the Atharva Veda, Sanat-Kumara-Samhita, Agni-Puraṇa, Brahmaṇḍa-Puraṇa, Brahma-Vaivarta-Puraṇa, and Kali-Santaraṇa-Upaniṣad, the recommendation is to chant Hare Krishna Mahamantra beginning with ‘Hare Krishna’. Exception is Padma Purana because it mentions ‘Hare Rama’ first, however there's no difference as already discussed above.
Jai Jai Shri Radhe!
Hare Krishna!
P.S. This article is an excerpt from the article The Four Sampradaya & Purpose of Sampradaya.