Just a few things I though I would chip into this discussion.
1) Syl say she had been following Kaladin around since the war, and Kaladin was surprised, implying he hadn't noticed.
EDIT QUOTE: Chapter 11 Droplets: "I watched you, you know. Back in that army."
2) Innate Soulcasting is just the Surge of Transformation, implying that Shallon has already formed a Nahel bond, In order to perform said Surge.
Taking those things into account I see two possibilities. Either the symbolhead-spren is Shallans Nahel-Spren, (Tarvagians(sp?) may be related to Void-binding, a dark match for Shallan like Szeth is for Kaladin), and the bond was formed earlier, and the Spren developed a sense of cognitive ability during that time before Shallan noticed (quite basic in my opinion all it wants is a truth, which due to its nature could be seen as natural instinct, earlier in its development than Syl who has a personality etc). This can be compared to Syl's development, gaining cognitive ability from Kaladin while Kaladin believes her to be a simple wind-spren, although he has the advantage of actually being able to see Syl.
OR the symbol head is not her Nahel-Spren and she has been bonded by her Nahel-Spren but simply hasn't noticed (unlikely).
Since Jasnah also has the Surge of Transformation, but does not recognize the Spren, it would imply and pretty much confirm that she is in a different order, with a Nahel-bond to a different type of Spren, again adding to the agprgument of the symbol-Spren being Shallan's Order's Nahel-Spren.
Anyway that was longer than I intended it to be, and a little confusing so sorry, added bits in back and forth as I went.
EDIT 2: just found another quote so going to make this even longer
Chapter72 Veristitalian : "These are a type of spren, Shallan. They are related to what you do.” She tapped the desk softly. “Two orders of the Knights Radiant possessed inherent Soulcasting ability; it was based on their powers that the original fabrials were designed, I believe.I had assumed that you… But no, that obviously wouldn’t make sense. I see now.”
“What?”
“I will explain as I train you,”"
I believe the bolded sentence, is where Jasnah realizes Shallan is of the other order, as she says it directly after mentioning two orders per Surge. Her only evidence as far as I can tell, is the type of spren that Shallan sees, which is also directly addressed just before she says it: "They are related to what you do"? This again implies Jasnah sees other spren than the symbol-heads.
Anyway, I'll let someone else have a go now.