Standard involutions conjugate to the negative of a standard involution in a Coxeter group Consider a finite irreducible Coxeter group $W$ with a fixed generator set $S$. Every involution in $W$ is conjugate to a standard involution $c_I$, for some subset $I\subset S$.  For example, this standard involution can be defined as the element $−1$ in a parabolic subgroup $W_I$; see [Ka01, §27], [Hu90, §8.3] or this question.  Moreover, there are explicit algorithms for determining conjugacy among these standard involutions.
Supposing that $-1 \in W$ (which is true for most types), we also have an involution $-c_I$ for any subset $I\subset S$.  My question is:  given $I$, how can we determine the subset $J$ for which $c_J$ is conjugate to $-c_I$?
It's not possible to determine this based on dimensions of eigenspaces alone.  For example, for $W$ of type $B_5$ or $C_5$, it's unclear to me which of the (non-conjugate) involutions $-c_{\{1\}}$ and $-c_{\{5\}}$ is conjugate to $c_{\{1,3,4,5\}}$ and which is conjugate to $c_{\{2,3,4,5\}}$.
[Ka01] Richard Kane, Reflection Groups and Invariant Theory
[Hu90] James E. Humphreys, Reflection Groups and Coxeter groups
 A: Partial answer
Concerning the particular example in the question, $-c_{\{1\}}$ is conjugate to $c_{\{1,3,4,5\}}$, and $-c_{\{5\}}$ is conjugate to $-c_{\{1,2,3,4\}}$.  This can almost be deduced from the dimension of the -1-eigenspaces, but with the following twist.  The Dynkin diagram of type $A_{10}$ “folds” onto the Dynkin diagram of type $C_5$.  This “folding” induces a surjection of corresponding sets of simple roots
$
\pi\colon \Sigma_{A_{10}} \twoheadrightarrow \Sigma_{B_5}
$
and an associated embedding
$$
\iota\colon  W(C_{5}) \hookrightarrow  W(A_{10})
$$
of the Weyl group of type $C_5$ into the Weyl group of type $A_{10}$.  (See this question.) The embedding sends $-1\in W(C_{5})$ to the longest element $w_0\in W(A_{10})$, and its image is the subgroup $ W_0 \subset W(A_{10})$ of elements that commute with $w_0$.  More generally:
Claim 1: The embedding $\iota$ sends the longest word $w_I = c_I$ of the parabolic subgroup associated with a subset of simple roots $I\subset \Sigma_{B_5}$ to the longest word $w_{\pi^{-1}I}$ of the parabolic subgroup corresponding to the "unfolded" subset $\pi^{-1}I\subset \Sigma_{A_{10}}$.
We thus obtain a very explicit bijection between $\mathcal C := $ {conjugacy classes of involutions in $W(C_{5})$} and $\mathcal A := $ {$W_0$-conjugacy classes of involutions in $W_0$}:

Under this bijection, multiplication with $-1$ on $\mathcal C$ corresponds to multiplication with $w_0$ on $\mathcal A$.
Now let $l^-(w)$ denote the dimension of the -1-eigenspace of $w$. On the representatives of involutions $w$ listed above, the tuple $((l^-(w),l^-(\iota w))$ takes the following values:
$(0,0)$,
$(1,2)$,
$(1,1)$,
$(2,4)$,
$(2,3)$,
$(2,2)$,
$(3,5)$,
$(3,4)$,
$(3,3)$,
$(4,4)$,
$(4,3)$,
$(5,5)$.
These are all distinct!  Therefore, $-w$ (for $w\in \mathcal C$) and $w_0w$ (for $w\in\mathcal A$) are uniquely determined by the tuple $(l^-(-w),l^-(w_0w))$.
Claim 2: By computing the tuples $(l^-(-w),l^-(w_0w))$, we find that:

The second and fourth line answer the question.
Generalizations
It is straightforward to extend this analysis to $W(B_n) = W(C_n)$ for arbitrary $n$, and to $W(F_4)$ (using the folding of $E_6$ onto $F_4$).  With some more care, the conjugacy class of $-c_I$ can also be computed for almost all $I$ for $W(D_{2n})$, but with a remaining ambiguity concerning the involutions corresponding to:

I do not know how to treat $W(E_7)$ and $W(E_8)$.
Details
Claim 1 follows from the explicit description of foldings in the first section of Steinberg, Endomorphisms of Linear Algebraic Groups (1968); see in particular 1.12(c), 1.32(a) and Corollary 1.33.  I would be interested whether there is a more direct, group theoretic argument for this claim.
For the calculation of $l^-(w_0w_I)$ that is implicit in claim 2, decompose the ambient vector space into the span of $I$ and its orthogonal complement, and observe that the restriction of $w_I$ to the complement is trivial.
A: Here is another approach to identify the conjugay class of the specific involution $-c_{\{1\}}$ in the question.   The approach is complementary to my other answer in that it also allows the determination of the conjugacy classes of $-c$ for all involutions in $W(E_7)$ and in $W(E_8)$.
There are precisely two conjugacy classes of involutions in $W(C_5)$ whose $-1$-eigenspace is one-dimensional.  These are represented by:

Likewise, there are precisely two conjugacy classe of involutions whose $-1$-eigenspace is four-dimenensional, and these are represented by:

The question is whether $-c_{\{1\}} \sim c_{\{1,3,4,5\}}$ and $-c_{\{5\}} \sim c_{\{2,3,4,5\}}$, or whether conversely $-c_{\{1\}} \sim c_{\{2,3,4,5\}}$ and $-c_{\{5\}} \sim c_{\{1,3,4,5\}}$.
As it turns out, for the approach I'm about to outline, it's best to start with $c_{\{5\}}$.  The fixed-point space of $c_{\{5\}}$ contains the sub-root system generated by the simple roots $\alpha_1,\alpha_2,\alpha_3$, since all of these are perpendicular to $\alpha_5$.  So the fixed-point space of $c_{\{5\}}$ contains a closed sub-root system of type $A_3$.  Equivalently, the $-1$-eigenspace of $-c_{\{5\}}$ contains a sub-root system of type $A_3$, and hence so does the $-1$-eigenspace of any conjugate of $-c_{\{5\}}$.
Now consider the two candidate standard representatives of the conjugacy class of $-c_{\{5\}}$ from $(*)$ above.  The $-1$-eigenspace of the first candidate is spanned by a closed sub-root system of type $A_1\times C_3$.  The classification of closed sub-root system of Borel and de Siebenthal implies that this sub-root system does not contain any sub-root system of type $A_3$.  (The maximal closed sub-root systems of a root system of type $C_3$ are of types $ A_1\times C_2$ and $A_2$.)  So the possiblity $-c_{\{5\}} \sim c_{\{1,3,4,5\}}$ can be excluded, and the correct answer must be:
$$
- c_{\{5\}} \sim c_{\{2,3,4,5\}}
$$
$$
- c_{\{1\}} \sim c_{\{1,3,4,5\}}
$$
Similar arguments resolve the only ambiguities in $E_7$ and $E_8$.  In $E_7$, we find that

as the $-1$-eigenspaces of the two involutions in the top line contain a sub-root system of type $A_2$, while the $-1$-eigenspace of the involution on the lower right cannot.  In $E_8$, we find that

as the $-1$-eigenspaces of $-c$ and $c$ contain a sub-root system of type $A_2$, while the $-1$-eigenspace of $c'$ does not.
