Shulmu and welcome to the community, u/tiawouldntwannabeeya.
The name of the goddess is Inana in Sumer and Akkad, and Ishtar in Assyria, and Babylonia. The most common written form is: 𒀭𒈹, AN.MUŠ₃ = diĝir-inana.
The etymology of Inana's name is uncertain, but by the Old Babylonian Period, ca. 1800 BCE, a folk-etymology had appeared that interpreted it as a contraction of: nin-an-na. How or why the name would have lost its initial ⟨n⟩ in the finalized form is uncertain.
This name is usually interpreted as: nin = "queen," an = "heaven," and na = the genitive "of," giving a meaning of: "Queen of Heaven," which agrees with her traditional role as the planet Venus. The people of Ancient Mesopotamia were aware of the planet's dual manifestation in the pre-dawn eastern, and post-dusk western sky as part of its heavenly circuit by the end of the fourth millennium BCE, when references to the "Morning Inana" (diĝir-inana-ḫud₂) and "Evening Inana" (diĝir-inana-sig) first appear in offering lists.
Alternatively, some Assyriologists have interpreted the name as: nin = "queen," an = "date-palm spadix," and na = the genitive "of," giving a meaning: "Queen of the Date-Palm Spadix," which also agrees with her role in the city of Uruk, where she was the tutelary goddess of the date-palm alongside her lover, Ama-ushumgal-ana, who was a local form of the god Dumuzi.
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With regards to statues, figurines, and votive pieces, there aren't many positively-identified representations of Inana or Ishtar in Mesopotamian artwork. Because of this, most modern-day polytheists make use of any figurine that "feels" Inana-esque to them, including replicas of the Burney Relief, which almost certainly represents Ishtar or another goddess named Ereshkigala, and not Lilith despite what Satanists and those who practice demonolatry or occultism might think.
Personally, I prefer to use replicas of the Mari Water-Pouring Goddess in my own devotional work. While this figurine was almost certainly not used to represent Inana or Ishtar in the ancient world, it includes some of the most important symbolism for a goddess from that time period: a horned-crown, a tiered robe, and a vase of life-giving water. The supplier, SacredSource, doesn't always have this replica in stock, so if you're called to it and they have one available I recommend purchasing it, because it might not be there when you look next time.
Amazon will also frequently sell stylized representations of Ishtar: here and here, for example. I often find these to be a little bit too "fantastical" for my liking, but I do admit to owning the second one linked above, and have seen many other Mesopotamian polytheists feature it on their altars as well. There are also an abundance of Burney Relief replicas for sale from all kinds of sellers, but these are usually mislabeled as Lilith figurines.
A variety of other sellers also exist that sometimes carry replicas related to Inana. Their pricing is often a bit steeper though, since they are hand-made many times. You can try searching for "Assyrian," "Babylonian" or "Mesopotamian" on Made4Museum's website, or check out DownUnderPharoah's Etsy and eBay stores for more.
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The process for awakening a cult statue in Ancient Mesopotamia involved the performance of two rituals: the "washing of the mouth" (mīs-pî) and the "opening of the mouth" (pīt-pî).
There is an elaborate ceremony that accompanies these rites. The most complete version of it can be explored via Michael B. Dick's excellent treatment, available: HERE
The goal of the ceremony was to put the statue into a state of ritual purity by removing any earthly dirt from it as well as scrubbing away any indication that human hands had been the ones to craft it.
To do this, the people of Ancient Mesopotamia used river water infused with herbs and precious metals to physically wash the statue. Then they lathered the mouth of the statue with holy oils to make it inviting to the God. Incense was lit and special prayers recited, the words carried to Heaven on the smoke where the ear of the God could hear them. The God would then descend to inhabit the statue. At the close of the ceremony the tools used to create the cultic statue were tossed into a river, its waters destroying them and erasing any sign that the statue had any origin other than its Heavenly conception.
Unfortunately, we don't know what all of the required ingredients are, nor have we recovered all of the required incantations that the priest was expected to recite. As a result, many of us perform our own versions of this rite, making use of our own ingredients and incantations.
Today, we use fresh water infused with our own plant matter to bathe the statue. The washing of the mouth follows, often performed with honey or juniper oil. The incense blends we use feature boxwood, cedar, cypress, frankincense, juniper, pine, myrrh, tamarisk, and other local species. Our prayers inviting the God to descend and dwell within the statue are either self-written, or curated from books and online catalogs of compositions. Finally, because we know that our versions are imitations of the real thing, some of us choose to re-dedicate our cultic icons once a month, although there is no rule requiring us to do so.
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