I am self-studying the 2nd edition of Casella and Berger's Statistical Inference and I'm working through the problems on exponential families. They define an exponential family of probability density functions as those that can be written in the form
\begin{equation*} f(x|\boldsymbol{\theta}) = h(x)c(\boldsymbol{\theta})\exp\left[\sum_{i=1}^{k}{w_{i}(\boldsymbol{\theta})t_{i}(x)}\right] \end{equation*}
in which $h(x)\geq 0$, the $t_{i}(x)$ are real-valued functions of the observation $x$ of the random variable $X$, $c(\boldsymbol{\theta})\geq 0$, and the $w_{i}(\boldsymbol{\theta})$ are real-valued functions of the parameter vector $\boldsymbol{\theta}$. One problem asks the reader to verify that the sub-family of normal probability density functions
\begin{equation*} f(x|\sigma^{2}) = \frac{1}{\sqrt{2\pi}\sigma}\exp\left[-\frac{1}{2\sigma^{2}}(x-\mu)^{2}\right] \end{equation*}
is an exponential family. I did this by writing $f(x|\sigma^{2})$ in the form of an exponential family with
\begin{align*} h(x) &= \frac{1}{\sqrt{2\pi}}\\ c(\sigma) &= \frac{1}{\sigma}I_{(0,\infty)}(\sigma)\\ w_{1}(\sigma) &= \frac{1}{\sigma^{2}}\\ t_{1}(x) &= -\frac{1}{2}(x-\mu)^{2}. \end{align*}
The solutions manual, on the other hand, puts
\begin{align*} h(x) &= 1\\ c(\sigma) &= \frac{1}{\sigma\sqrt{2\pi}}I_{(0,\infty)}(\sigma)\\ w_{1}(\sigma) &= -\frac{1}{2\sigma^{2}}\\ t_{1}(x) &= (x-\mu)^{2}. \end{align*}
At first, the apparently trivial differences between the my answer and the given solution seemed inconsequential, but the book goes on to discuss the natural parameter space of the family of probability density functions, which is defined as
\begin{equation*} \mathcal{H} = \left\{(\eta_{1},\ldots,\eta_{k})\in \mathbb{R}^{k}~;~\int_{-\infty}^{\infty}{h(x)\exp\left[\sum_{i=1}^{k}{\eta_{i}t_{i}(x)}\right]\mathrm{d}{x}}<\infty\right\} \end{equation*}
Using my answer, it seems like the natural parameter space would be $\mathcal{H} = \left\{\eta_{1}\in \mathbb{R}~;~\eta_{1}>0\right\}$, but using the solutions manual, it would be $\mathcal{H} = \left\{\eta_{1}\in \mathbb{R}~;~\eta_{1}<0\right\}$.
My instinct is that a "natural" parameter space should be unique (this might not be justified), but both of these seemingly correct solutions yield different natural parameter spaces. This brings me to my question: is there a convention that exists regarding whether the $w_{i}$ or the $t_{i}$ should carry negative signs or scale factors?
