Was there a symbol for negative one? This may be more of a historical math question.
In any old, ancient, or even modern symbolism; was there a symbol or letter that represented the value $-1$?
This goes in pair with how ancient mathematicians would commonly use a symbol for $0$ before the concept of zero was well defined.
 A: $-1$ has not had any special symbols in widespread use. It was "minus" that has had various symbols for its representation both to negate a quantity and/or as subtraction operation (some of them are $m^\circ$, meno, $m\tilde e$, $men$, $\tilde{m}$, $\sim$, $^{\cdot}\hspace{-6pt}{-}$, $\div$, and many others).
There are lots of books on this subject.
There is just one exception in what I've just said. In 1925 the mathematician J. P. Ballantine had a similar intuition as yours. In American Mathematical Monthly Vol. 32 No. 6 (Jun.-Jul. 1925) p.302 he wrote:

Mathematical historians will tell you how many years mathematics was held back for want of a digit $0$. Though not comparing in importance with that digit, there is a certain advantage in having a digit to represent negative one. 

He continues proposing the adoption for $-1$ of a symbol which is $1$ rotated by $180^\circ$ around its center (in mathjax its difficult to represent; it is something like $\downharpoonright$. In Latex it is much more simple using this syntax: \rotatebox[origin=c]{180}{1}, after having imported the graphics package with \usepackage{graphicx}) 
Please note that Ballantine proposes such a symbol not for the number $-1$ but for $-1$ as a digit.
In this way not only it is possible to write $${\downharpoonright}\times2=-2$$ but it is also possible to write: $${\downharpoonright}.123=-1+0.123$$ $${\downharpoonright}0=-10$$ $${\downharpoonright}2=-10+2$$ with $${\downharpoonright}2={\downharpoonright}92=...={\downharpoonright}99992=...$$
In the end Ballantine's proposal as himself says it is a way to represent numbers in their tens' complement representation.

EDIT:
Ballantine does not give an example of numbers where the digit $\downharpoonright$ is not the most significant digit, probably because he does not need such sofistication in his intended application (that is, to complement a given positive number $x$ w.r.t. an integer $n$-th power of ten, where $n\ge\log_{10} x$ ). Anyway following his philosophy we could also write:
$${\downharpoonright}{\downharpoonright}=-11$$
$$10.{\downharpoonright}=10-0.1$$
That said, Ballantine originality is mainly about the new symbol proposed for the digit $-1$. As to the applications it is not as much as original. Think that the periodical editor recalls in a footnote (Op. cit.) that a generalization coud be one in which digits $\bar5, \bar4, \bar3, \bar2, \bar1, 0,1,2,3,4,5$ are used instead of $0,1,2,3,4,5,6,7,8,9$ for real number encoding, where $\bar5=-5,...$
Indeed this was something very well known in past centuries. As early as 1726 John Colson in "A Short Account of Negativo-Affirmative Arithmetick" Philosophical Transactions of the Royal Society of London Vol 34 pp. 161-173 introduces such negative figures, as he calls them, as a means to ease arithmetic especially when large numbers are involved. It is no more than tens' complement encoding of numbers.
A: In balanced ternary, there is a symbol for negative one.
If T is negative one, then in balanced ternary, you count
0000,
0001,
001T,
0010,
0011,
01TT,
01T0,
01T1,
etc
