I was reading a bit about them yesterday, in the references included with this answer and this answer.
What the Buddha did not teach by Christopher Titmuss is I suppose Theravada, and/or perhaps his own interpretation based on the suttas.
In that article, section 4 describes "Belief in God", and "Brahma"; and then section 25 says,
- Oneness is the ultimate realty. The Buddha did not refute the importance of Oneness. He refers to it in the experience of Brahma Viharas (Divine Abode or Abode of God ). He does not refer to this Abode as ultimate reality since it posits the notion of a self that unites with other. The Buddha refutes All is One and equally refutes All is Many and points to dependent arising. In MLD 117, the Buddha said: Unification of mind equipped with the other seven links of the Eighfold Path is called noble right rconcentation [sic] with its supports and requisites.
What the Buddha did not teach also has a couple of sections (19 and 20) about Metta.
This description of Atammayata (see also Samadhi's answer which explains it) describes it as, I think, a necessary antidote to equanimity.
Why does Tan Ajarn Buddhadasa consider atammayata so important? In the Salayatanavibhanga Sutta of the Majjhima-nikaya (#137) the Buddha describes a spiritual progression carried out by "relying on this, to give up that." Relying on the pleasure, pain, and equanimity associated with renunciation, one gives up the pleasure and pain associated with worldliness. Relying on singular or one-pointed equanimity (ekaggata-upekkha), one gives up many-sided or multifaceted equanimity (nanatta-upekkha). Relying on atammayata, one gives up ekaggata-upekkha.
In this sutta, nanatta-upekkha is explained as "equanimity toward forms, sounds, odors, tastes, touches, and mind-objects," which implies the four meditative states known as the "rupa-jhana." Ekaggata-upekkha is explained as "equanimity dependent upon the four immaterial absorptions (arupa-jhana)." To more easily understand what this means, we may compare it with the common Buddhist hierarchy of the sensual (kama-), pure material (rupa-), and non-material (arupa-) realms. The ordinary worldling or "Thickster" (putthujhana) clings to sensual experiences due to craving for sensual pleasures. One gets free of sensuality by relying on pure materiality, that is, steady concentration upon material objects (rupa-jhana). Pure materiality is abandoned by relying on the arupa-jhana. Finally, these exalted states of consciousness are abandoned through atammayata.
Something like the Brahmavihara Sutta says,
"This awareness-release through equanimity (and/or through good-will) should be developed whether one is a woman or a man. Neither a woman nor a man can go taking this body along. Death, monks, is but a gap of a thought away. One [who practices this awareness-release] discerns, 'Whatever evil action has been done by this body born of action, that will all be experienced here [in this life]. It will not come to be hereafter.' Thus developed, the awareness-release through equanimity leads to non-returning for the monk who has gained gnosis here and has penetrated to no higher release."
There's an article on accesstoinsight titled The Four Sublime States –
Contemplations on Love, Compassion, Sympathetic Joy and Equanimity by Nyanaponika Thera (a German-born Sri-Lanka-ordained Theravada monk).
It includes (without referencing suttas) a dozen short paragraphs on each of the four states; and another dozen at the end describing inter-relations between states.
It says, near the beginning,
These four attitudes are said to be excellent or sublime because they are the right or ideal way of conduct towards living beings (sattesu samma patipatti). They provide, in fact, the answer to all situations arising from social contact.
When I read the question Why does insight lead to to compassion? I wondered whether there's ever "increasing levels of insight" without any "social contact": and decided it's unlikely, and that human existence (physical human livelihood) tends to depend on some social contact.
The non-Theravada probably have a lot more to say; for example the Dalai Lama says that people "always feel happy" from helping others, etc.