Let $X_1$ be a positive random variable with a continuous cumulative distribution function and $C$ the copula of $(X_1, 1/X_1, \exp(-X_1))$.
Why is $C$ also the copula of $(X_1, -X_1, -X_1)$ and what is the form of $C$?
For the first question, we have by Sklar's theorem that $C(X_1, 1/X_1, \exp(-X_1)) = F\big(q(X_1), q(1/X_1), q(\exp(-X))\big)$, where $q$ denotes the quantile function of $X$ which is just the inverse of the cdf of $X$ since the cdf of $X$ is continuous. So this would imply that we would need $q(1/X_1) = q(\exp(-X_1)) = q(-X_1)$, but I can't think of a distribution for which this would hold.
Further, for the second question regarding the form of $C$, I was thinking maybe using the independence copula $\Pi(\mathbf{u}) = \prod_{j=1}^d u_j$ since then the minus of the second and third component cancel each other out.
The answer follows from the comonotonicity/countermonotonicity of the variables considered. To see this, for simplicity consider a case with exact, continuous inverses. In this case we have by definition $$\begin{aligned}C(u_1,u_2,u_3)=P(X_1\leq F_{X_1}^{-1}(u_1),1/X_{1}\leq F_{1/X_1}^{-1}(u_2),e^{-X_1}\leq F_{e^{-X_1}}^{-1}(u_3)) \end{aligned}$$ However $$F_{1/X_1}(x_2)=P(X_1\geq 1/x_2)=1-F_{X_1}(1/x_2)\implies F_{1/X_1}^{-1}(u_2)=1/F_{X_1}^{-1}(1-u_2)$$ $$F_{e^{-X_1}}(x_3)=P(X_1\geq -\ln(x_3))=1-F_{X_1}(-\ln(x_3))\implies F_{e^{-X_1}}^{-1}(u_3)=e^{-F_{X_1}^{-1}(1-u_3)}$$ Therefore we get $$C(u_1,u_2,u_3)=P(X_1\leq F_{X_1}^{-1}(u_1),X_1\geq F_{X_1}^{-1}(1-u_2),X_1\geq F_{X_1}^{-1}(1-u_3))$$ Now consider $$\tilde{C}(u_1,u_2,u_3)=P(X_1\leq F_{X_1}^{-1}(u_1),-X_{1}\leq F_{-X_1}^{-1}(u_2),{-X_1}\leq F_{{-X_1}}^{-1}(u_3))$$ We have $$F_{-X_1}(x_2)=P(X_1\geq -x_2)=1-F_{X_1}(-x_2)\implies F_{-X_1}^{-1}(u_2)=-F_{X_1}^{-1}(1-u_2)$$ So ${C}(u_1,u_2,u_3)=\tilde{C}(u_1,u_2,u_3)$ and we conclude for the first part. For the form: $$\begin{aligned}{C}(u_1,u_2,u_3)&=P(\max(F_{X_1}^{-1}(1-u_2),F_{X_1}^{-1}(1-u_3))\leq X_1\leq F_{X_1}^{-1}(u_1))=\\ &=F_{X_1}(F_{X_1}^{-1}(u_1))-F_{X_1}(\max(F_{X_1}^{-1}(1-u_2),F_{X_1}^{-1}(1-u_3))=\\ &=u_1-\max(1-u_2,1-u_3),\quad u_1\geq \max(1-u_2,1-u_3)\end{aligned}$$ and $0$ otherwise.