Given $\Omega \subseteq \Bbb{R}^N$, and an $x'$-symmetric domain $x=(x', x'')$ with $x' \in \Bbb{R}^{k}$ and $x'' \in \Bbb{R}^{N-k}$, the authors first prove that for every $a \in (0, k-2]$, there exists a constant $C\in (0, \infty)$ such that if the set $\Omega$ is $x'$-symmetric and has finite measure, if the function $u \in C_{0}^{\infty}(\Omega)$ is an $x'$-symmetric function and if $$ \int_{\Omega}|\nabla u(x)|^{N-a} |x'|^{-a} \,\roman{d} x \leq 1, $$ then $$ \frac{1}{|\Omega|} \int_{\Omega} \roman{e}^{\beta_{a}|u|^{\frac{N-a}{N-a-1}}} |x'|^{-a} \,\roman{d} x \leq C $$ with the constant $$ \beta_{a}=(N-a)\biggl(\frac{2 \pi^{\frac{N}{2}} \Gamma\bigl(\frac{k-a}{2}\bigr)}{\Gamma\bigl(\frac{k}{2}\bigr) \Gamma\bigl(\frac{N-a}{2}\bigr)}\biggr)^{\frac{1}{N-a-1}}, $$ which cannot be replaced by a larger value. Similarly, there exists $C \in (0, \infty)$ such that if $u \in C_{0}^{\infty}(\Bbb{R}^{N})$ is $x'$-symmetric, if $\tau > 0$ and if $$ \int_{\Bbb{R}^{N}}\left(|\nabla u|^{N-a}+\tau|u|^{N-a}\right)|x'|^{-a} \,\roman{d} x \leq 1, $$ then $$ \int_{\Bbb{R}^{N}}\biggl(\exp \left(\beta_{a}|u|^{\frac{N-a}{N-a-1}}\right)-\sum_{k=0}^{\lceil N - a - 2\rceil} \frac{\beta_{a}^{k}|u|^{\frac{k(N-a)}{N-a-1}}}{k !}\biggr)\left|x'\right|^{-a} \,\roman{d} x \leq C, $$ where $\lceil t\rceil =\min \{j \in \Bbb{Z}: j \geq t\}$. \par As an application, they obtain weighted Trudinger-Moser inequalities for Grushin operators: Let $\alpha \geq 2-m-\gamma n$. Set $$ A(\alpha)=(1+\gamma)\left(Q_{\gamma}+\alpha\right)\biggl(\frac{2 \pi^{\frac{n+m}{2}} \Gamma\left(\frac{m+\gamma n+\alpha}{2}\right)}{\Gamma\left(\frac{m}{2}\right) \Gamma\bigl(\frac{Q_{\gamma}+\alpha}{2}\bigr)}\biggr)^{\frac{1}{Q_{\gamma}+\alpha-1}}, $$ where $Q_{\gamma}=m+(1+\gamma) n$. Then there exists a positive constant $C>0$ such that if $D \subset \Bbb{R}^{m+n}$ is an $x$-symmetric domain with finite measure, if the function $u \in C_{0}^{\infty}(D)$ is $x$-symmetric and if $$ \int_{D}\left|\nabla_{G} u\right|^{Q_{\gamma}+\alpha}|x|^{\alpha} \,\roman{d} x \,\roman{d} y \leq 1, $$ where $\nabla_{G}=(\nabla_{x},(1+\gamma) \nabla_{y})$, then $$ \frac{1}{\int_{D}|x|^{\gamma} Q_{\gamma}+(1+\gamma) \alpha \,\roman{d} x \,\roman{d} y} \int_{D} \exp \Bigl(A(\alpha)|u|^{\frac{Q_{\gamma}+\alpha}{Q_{\gamma}+\alpha-1}}\Bigr)|x|^{\gamma Q_{\gamma}+(1+\gamma) \alpha} \,\roman{d} x \,\roman{d} y \leq C, $$ the constant $A(\alpha)$ being sharp; if $u \in C_{0}^{\infty}(\Bbb{R}^{m+n})$ is $x$-symmetric, if $\tau > 0$ and if $$ \int_{\Bbb{R}^{m+n}}\left|\nabla_{G} u\right|^{Q_{\gamma}+\alpha}|x|^{\alpha} +\tau |u|^{Q_{\gamma}+\alpha}|x|^{\gamma Q_{\gamma}+(1+\gamma) \alpha} \,\roman{d} x \, \roman{d} y \leq 1 $$ then $$ \int_{\Bbb{R}^{m+n}}\biggl(\exp \Bigl(A(\alpha)|u|^{\frac{Q_{\gamma}+\alpha}{Q_{\gamma}+\alpha-1}}\Bigr)-\sum_{k=0}^{\lceil Q_\gamma + \alpha - 2\rceil} \frac{A(\alpha)^{k}|u|^{\frac{k\left(Q_{\gamma}+\alpha\right)}{Q_{\gamma}+\alpha-1}}}{k !}\biggr)|x|^{\gamma Q_{\gamma}+(1+\gamma) \alpha} \, \roman{d} x\,\roman{d}y \leq C. $$ \par If $\Bbb{G} \simeq \Bbb{R}^{m + n}$ is a group of type $H$ in the sense of Kaplan, setting for $a \geq 2-m-n$ $$ A_{a}(\Bbb{G})=2(Q+a)\biggl(\frac{2 \pi^{\frac{m+n}{2}} \Gamma\bigl(\frac{m+n+a}{2}\bigr)}{4^{n} \Gamma\bigl(\frac{m}{2}\bigr) \Gamma\bigl(\frac{Q+a}{2}\bigr)}\biggr)^{\frac{1}{Q+a-1}} $$ where $Q$ is the homogeneous dimension of $\Bbb{G}$, there exists a positive constant $C>0$ such that if $\Omega \subset \Bbb{R}^{m+n}$ is an $x$-symmetric domain and has finite area, if $F \in C_{0}^{\infty}(\Omega)$ is an $x$-symmetric function and if $$ \int_{\Omega}\left|\nabla_{G} F\right|^{Q+a}|x|^{a} \,\roman{d} x \,\roman{d} y \leq 1, $$ then $$ \frac{1}{\int_{\Omega}|x|^{Q+2 a} \,\roman{d} x\,\roman{ d } y} \int_{\Omega} \exp \Bigl(A_{a}(\Bbb{G})|F|^\frac{Q+a}{Q+a-1}\Bigr)|x|^{Q+2 a} \,\roman{d} x \,\roman{d} y \leq C ; $$ if $F \in C_{0}^{\infty}(\Bbb{R}^{m+n})$ is an $x$-symmetric, if $\tau > 0$ and if $$ \int_{\Bbb{R}^{m+n}}\left|\nabla_{G} F\right|^{Q+a}|x|^{a} +\tau |F|^{Q-a}|x|^{Q+2 a}\,\roman{d} x \,\roman{d} y \leq 1, $$ then $$ \int_{\Bbb{R}^{m+n}}\biggl(\exp \Bigl(A_{a}(\Bbb{G})|F|^\frac{Q+a}{Q+a-1}\Bigr)-\sum_{k=0}^{\lceil Q + a -2\rceil} \frac{A_{a}(\Bbb{G})^{k}|F|^{\frac{k(Q+a)}{Q+a-1}}}{k !}\biggr)|x|^{Q+2 a} \,\roman{d} x \,\roman{d} y \leq C. $$