An ancient narrative
One of the oldest artifacts in the world relating to the sex Goddess Inanna is known as the Uruk Vase, or Warka Vase, showing the arrival of the King-God. It is one of the earliest pieces of narrative sculpture in human history, dating to around 3200 BCE, making it over 5000 years old.
Uruk vase or Warka vase

It features a middle frieze of naked men bringing offerings of fruit and grain to the Goddess (and / or her High Priestess), while the top register shows the King arriving. It's entirely possible that this very artifact depicts the quršu sex ritual.
Uruk vase
A Goddess with many lovers
In mythology, Inanna's consort was the shepherd-King named Dumuzi (also Tammuz in later Akkadian). 

From archaeological King Lists, there was a historical king (in fact several figures) named Dumuzi, upon whom the myth may have been originally based. It appears that over time, the King or a selected high priest played the role of Dumuzi in the theatrical ritual performance of the quršu.

The Goddess Inanna / Ishtar had a reputation for turning on any lover that crossed her, or who proved himself disloyal or unworthy. (15) 

Based on the archaeological evidence, I believe that in some places and certain eras, the quršu ritual was enacted out with human protagonists playing the roles of Inanna and the King-God, (such as by a high priestess representing the Goddess and the reigning King representing the God), while in other places it may have been an imagined ritual where the King-God went to the chamber of the Goddess within her temple and she was imagined to make a night visit.
For example, in the ancient text, Enmerkar and Ensuhgirana, two rival Kings from different cities are debating who has Inanna's affection. Enkemar boasts of his superior sexual connection with the Goddess:
"he may lie with her on the splendid bed, but I lie in sweet slumber with her on the adorned bed, he may see dreams with Inanna at night, but I converse (IMIN-BAL) with Inanna awake." (16)

The term "imin-bal" is formed by the word IMIN for talk, with BAL - the same term earlier referred to with the association of the loom-shuttle and reversing. It can be translated as converse with talk back-and-forth, or talk with erotic connotations, talking with genitals if you will. The Goddess's support of a King ensures his domination over others.

In some of the sex songs, Inanna chooses her lover, from several suitors. In other songs, the inherent qualities of the chosen suitor mean that he is worthy of Inanna's vulva.
Bringing the Lady what she desires
The King-God Dumuzi (or the King playing her in a particular historical reign) brings Goddess Inanna "what she desires":
"...whatever your heart desires, I brought,
Whatever your heart, your loving heart desires, I brought....
Shining sister, to whom I have brought five things,
My sister to whom I have brought ten things..." (17)

Among the things brought which are alluded to within the song are lambs, ewes and goats; which align with the imagery of the Uruk vase. Tracing the mythological roots of Inanna and Dumuzi, they were Goddess of the sheepfold and sheperd-God respectively, both implicated in libido, vigour, fertility and abundance.

In some of the sex songs there is reference to cheese, cream and milk products, likely chosen ritually as products of sheep and goats, but also alluding to the sexual act and resemblance of the sexual fluids.
"Oh that I might know the way to Amasumgalanna [the King], my milk, my cream!" (18)

"When he carried the butter, butter multiplied,
When he carried the milk, milk multiplied...[...]
Oh that I might know the way to the bridegroom [Dumuzi[, my milk! my cream!" (19)

I find this very interesting also to relate in the contemporary culture practice of a lover buying chocolate for a lady on romantic occasions (such as for Valentines Day) - with chocolate itself being a milk product.

In other sex songs, jewellery and semi-precious stones are brought as a gift:
"The man has indeed brought the maid Inanna, has indeed brought a heap of precious stones to choose from." (20)
One of the most highly valued stones in Mesopotamia was lapis lazuli, the beautiful blue stone with silvery-gold inflections, resembling the starry night sky. 
She arranges the various beads and jewels on her buttocks, nape, hair, ear lobes, naval, hips, thigh and vulva. (21)
There are stunning examples of the range of jewellery excavated from the so-called Royal Tombs of Ur, worn by Nin Puabi and her attendants or priestesses. The jewellery featured rich assemblages of lapis lazuli with carnelian and gold, travelling vast distances along the lapis lazuli route from the Indus Valley to the mines of Afghanistan and into Southern Iraq.
Other pieces of jewellery were fashioned across the Levantine Coast and into the Mediterranean and Cyprus, gifted predominantly to women in courting and marriage in which the Quršu sex songs of the Goddess and God are likely to have been sung. Just as in a similar way, the biblical Song of Songs is one of the most common passages read from in wedding ceremonies, which itself follows the tradition of the ancient BAL-BAL-E sex songs.
(15) In the Epic of Gilgamesh, the King Gilgamesh after returning from the Cedar Forest rejects the amorous desires by the Goddess Ishtar (Inanna), listing the fate which has befallen her previous lovers. see for example Sandars, N. K (translator) Epic of Gilgamesh (1960) Penguin Books, p.84. It should also be noted that in the Descent of Inanna to the Underground, she finds Dumuzi sitting on her throne and not grieving her death, and determines that he will take her place in the Underworld.
(16) "Enmerkar and Ensuḫgirana" The Electronic Text Corpus of Sumerian Literature t. viewable at: (See passage centre within lines 25-39)
(17) Sefati, Yitschak Love Songs in Sumerian Literature (1998) Bar Ilan University Press, see Dumuzi-Inanna C lines 19-24 p.136
(18) Sefati, Yitschak Love Songs in Sumerian Literature (1998) Dumuzi-Inanna R  Source A, line 10, p.238
(19) Sefati, Yitschak Love Songs in Sumerian Literature (1998) Dumuzi-Inanna R  Source C, lines 4-8, p.239
(20) Sefati, Yitschak Love Songs in Sumerian Literature (1998) Dumuzi-Inanna T, line 8, p.249
(21) Sefati, Yitschak Love Songs in Sumerian Literature (1998) Dumuzi-Inanna T, see lines 11-23, p.249-250