One thing he argues is that Sūrat al-Fātiḥa is a ring composition. In Muslim faith and practice, the Fātiḥa (literally meaning, "The Opening") is not only the first sūra ("chapter," for lack of a better term) in the Qur'an, but it is also the perfect prayer taught to them by God to be recited as the most important part of the daily prayers, and also represents a summary of the main themes of the Qur'an.
I agree with Farrin's observation about the sūra's chiastic structure, but I have a slightly different take on the explanation of the relationship between the different terms in this structure. Here I will diagram the structure of the sūra and then explain what I believe is its precise import.
الحمد للهA. رب العالمينB. الرحمنB'. الرحيمA'. مالك يوم الدينC. إياك نعبدC'. وإياك نستعينA. اهدنا الصراط المستقيمB. صراط الذين أنعمت عليهمB'. غير المغضوب عليهمA'. ولا الضالين
All praise are due to GodA. The Lord of all peoples,B. The All-Merciful,B'. The Ever-Merciful,A'. Master of the Day of Recompense.C. You alone we worship,C'. and you alone we ask for help.A. Guide us along the Straight Path,B. the path of those whom You have favored,B'. not of those who have earned wrath,A'. nor of the astray.
As Farrin points out, this sūra is composed of two chiasmi fit into a ring composition. The structure gives profound insight into the meaning of the individual verses of the sūra. However, the following is what I believe to be a more accurate account of the meaning within the two chiasmi.
Let me begin by explaining the first chiasmus (in blue):
A describes God's relationship with people in this world, while A' describes his relationship with them in the afterlife. A uses the term rabb (translated as "Lord") while A' uses the term mālik (translated as "Master"). These are synonyms in the Arabic language, mālik being one of the meanings of the word rabb.
The relationship between B and B' is obvious. They are both names of God highlighting his mercy (raḥma). However the nuance in their meanings is brought out by their placement in the chiasmus. B is closely tied to A, because they primarily concern this world. B' is closely tied to A' because they primarily concern the next world.
Let us skip C-C' for a moment and look at the second chiasmus (in red):
The relationship between A and B is obvious, and the relationship between B' and A' is obvious. The former concern those who are guided, while the latter concern those who are not.
What is really interesting is the relationship between A and A', and B and B'. A concerns guidance (hidāya) while A' concerns its antithesis, misguidance/going astray (ḍalāl). B concerns a class of people labeled alladhīna anʿamta ʿalayhim ("those whom You have favored"), while B' concerns a class of people labeled al-maghḍūbi ʿalayhim ("those who have earned wrath"). These are not exact antithesis, but this appears to be an example of a literary device the Qur'an uses in which it juxtaposes two not-quite opposite terms in order to create a four-way comparison. The antithesis of niʿmah ("favor" - B) in Arabic is niqma ("retribution, punishment"), while the antithesis of ghaḍab ("wrath" - B') is riḍā ("satisfaction"). The implication is that those who are qualified of God's favor (niʿmah) are also qualified by His satisfaction (riḍā), while those who have brought on His wrath (ghaḍab) have also brought on His retribution (niqma).
Now returning to the center (C-C'):
"You alone we worship and You alone we ask for help" is the central idea of this sūrah. By implication, since this sūrah summarizes the contents of the Qur'an, the verse embodies the main idea of the entire Qur'an. It is the relationship of the worshiper (ʿābid) with his God (ilāh), and of the slave (ʿabd) with his Lord (rabb). It is equivalent to the Muslim declaration of faith, "there is nothing worthy of worship except God" (lā illāha illā 'llāh).
C concerns the exclusive worship of God. Now look at the entire first half of the sūra ending with C—from "All praise are due to God" until "You alone we worship." The whole first half in fact consists of exactly that: worship of God.
C' concerns praying exclusively to God for help. Now look at the entire second of the sūra beginning with C'—from "You alone we ask for help" until "nor of the astray." The whole second half of the sūra actually consists entirely of praying to God for help!
So the sūra has an extremely brilliant structure which is closely tied to its meanings. It is so precisely worded and arranged that nothing could be added or taken away from it without breaking the entire composition. Moreover it beautifully summarizes the main themes of the Qur'an.
There is a lot more that can be said about the precise wording of the sūra, how the verses connect linearly, and other features of its naẓm (coherence and arrangement). I do not intend to discuss all of these here. However I would like to devote the next post to a few more observations about the implication of the structure of this sūrah, shown above, on its meanings as well as how it connects with the final sūra of the Qur'an. In shā’a 'llāh.
 There is a lot of commentary on the precise distinctions between two names. To summarize what is relevant here, it can be said that ar-Raḥmān (which I have translated as "the All-Merciful") connotes God's mercy towards all of His creatures, deserving and undeserving alike; while ar-Raḥīm highlights the permanence of God's mercy, but which is more selective and excludes some types of people in the afterlife.
 The bipartite structure of this sūra is alluded to in a famous ḥadīth qudṣī ("sacred tradition") in which the Prophet related on behalf of God "I have divided the prayer [i.e. Sūrat al-Fātiḥa] into two halves between Myself and My slave, and My slave shall have what he asks for."