No proofs, just experimental facts (hence cw):

Let $a_{n+1}=xa_n+\varepsilon_{n+1}a_{n-1}$, with $\varepsilon_k=\pm1$; then $\varepsilon_{2^k}$ might be arbitrary, while all others are uniquely determined by them. For the first few cases we have $$ \begin{aligned} \varepsilon_{3}&=-\varepsilon_{2}\\ \varepsilon_{5}&=-\varepsilon_{4}\\ \varepsilon_{6}&=-\varepsilon_{2}\\ \varepsilon_{7}&=\varepsilon_{2}\\ \varepsilon_{9}&=-\varepsilon_{8}\\ \varepsilon_{10}&=\varepsilon_{2}\\ \varepsilon_{11}&=-\varepsilon_{2}\\ \varepsilon_{12}&=-\varepsilon_{4}\\ \varepsilon_{13}&=\varepsilon_{4}\\ \varepsilon_{14}&=-\varepsilon_{2}\\ \varepsilon_{15}&=\varepsilon_{2}\\ \varepsilon_{17}&=-\varepsilon_{16}\\ \varepsilon_{18}&=\varepsilon_{2}\\ \varepsilon_{19}&=-\varepsilon_{2}\\ \varepsilon_{20}&=\varepsilon_{4}\\ \varepsilon_{21}&=-\varepsilon_{4}, \end{aligned} $$ the polynomials being, respectively, $$ \begin{aligned} a_1&=x \\ a_2&=x^2+\varepsilon_{2} \\ a_3&=x^3 \\ a_4&=x^4+\varepsilon_{4}x^2 +\varepsilon_{2} \varepsilon_{4} \\ a_5&=x^5+x \varepsilon_{2} \varepsilon_{4} \\ a_6&=x^6-\varepsilon_{2}x^4 -\varepsilon_{4} \\ a_7&=x^7 \\ a_8&=x^8+\varepsilon_{8}x^6 -\varepsilon_{2} \varepsilon_{8}x^4 -\varepsilon_{4} \varepsilon_{8} \\ a_9&=x^9-\varepsilon_{2} \varepsilon_{8}x^5 -\varepsilon_{4} \varepsilon_{8}x \\ a_{10}&=x^{10}+\varepsilon_{2}x^8 -\varepsilon_{8}x^4 - \varepsilon_{4} \varepsilon_{8}x^2-\varepsilon_{2}\varepsilon_{4} \varepsilon_{8} \\ a_{11}&=x^{11}-\varepsilon_{4} \varepsilon_{8}x^3 \\ a_{12}&=x^{12}-\varepsilon_{4}x^{10} -\varepsilon_{2} \varepsilon_{4}x^8 + \varepsilon_{8}x^2+\varepsilon_{2}\varepsilon_{8} \\ a_{13}&=x^{13}-\varepsilon_{2} \varepsilon_{4}x^9 +\varepsilon_{2} \varepsilon_{8}x \\ a_{14}&=x^{14}-\varepsilon_{2}x^{12} +\varepsilon_{4}x^8 - \varepsilon_{8} \\ a_{15}&=x^{15} \\ a_{16}&=x^{16}+\varepsilon_{16}x^{14} -\varepsilon_{2} \varepsilon_{16}x^{12} +\varepsilon_{4} \varepsilon_{16}x^8 -\varepsilon_{8} \varepsilon_{16} \\ a_{17}&=x^{17}-\varepsilon_{2} \varepsilon_{16}x^{13} + \varepsilon_{4} \varepsilon_{16}x^9 - \varepsilon_{8} \varepsilon_{16}x \\ a_{18}&=x^{18}+\varepsilon_{2}x^{16} -\varepsilon_{16}x^{12} +\varepsilon _{4} \varepsilon_{16}x^{10} +\varepsilon_{2}\varepsilon_{4} \varepsilon_{16}x^8 - \varepsilon_{8} \varepsilon_{16}x^2 -\varepsilon_{2} \varepsilon_{8} \varepsilon_{16} \\ a_{19}&=x^{19}+\varepsilon_{4} \varepsilon_{16}x^{11} - \varepsilon_{8} \varepsilon_{16}x^3 \\ a_{20}&=x^{20}+\varepsilon_{4}x^{18} +\varepsilon_{2} \varepsilon_{4}x^{16} +\varepsilon_{16}x^{10} +\varepsilon_{2}\varepsilon_{16}x^8 -\varepsilon_{8} \varepsilon_{16}x^4 -\varepsilon_{2}^4 \varepsilon_{4}^3 \varepsilon_{8} \varepsilon _{16}x^2 -\varepsilon_{2}\varepsilon_{4}\varepsilon_{8} \varepsilon_{16} \\ a_{21}&=x^{21}+\varepsilon_{2} \varepsilon_{4}x^{17} +\varepsilon_{2} \varepsilon_{16}x^9 - \varepsilon_{8} \varepsilon_{16}x^5- \varepsilon_{2} \varepsilon_{4} \varepsilon_{8} \varepsilon_{16}x \end{aligned} $$

No proofs, just experimental facts (hence cw):

Let $a_{n+1}=xa_n+\varepsilon_{n+1}a_{n-1}$, with $\varepsilon_k=\pm1$; then $\varepsilon_{2^k}$ might be arbitrary, while all others are uniquely determined by them. For the first few cases we have $$ \begin{aligned} \varepsilon_{3}&=-\varepsilon_{2}\\ \varepsilon_{5}&=-\varepsilon_{4}\\ \varepsilon_{6}&=-\varepsilon_{2}\\ \varepsilon_{7}&=\varepsilon_{2}\\ \varepsilon_{9}&=-\varepsilon_{8}\\ \varepsilon_{10}&=\varepsilon_{2}\\ \varepsilon_{11}&=-\varepsilon_{2}\\ \varepsilon_{12}&=-\varepsilon_{4}\\ \varepsilon_{13}&=\varepsilon_{4}\\ \varepsilon_{14}&=-\varepsilon_{2}\\ \varepsilon_{15}&=\varepsilon_{2}\\ \varepsilon_{17}&=-\varepsilon_{16}\\ \varepsilon_{18}&=\varepsilon_{2}\\ \varepsilon_{19}&=-\varepsilon_{2}\\ \varepsilon_{20}&=\varepsilon_{4}\\ \varepsilon_{21}&=-\varepsilon_{4}, \end{aligned} $$ the polynomials being, respectively, $$ \begin{aligned} a_1&=x \\ a_2&=x^2+\varepsilon_{2} \\ a_3&=x^3 \\ a_4&=x^4+\varepsilon_{4}x^2 +\varepsilon_{2} \varepsilon_{4} \\ a_5&=x^5+\varepsilon_{2} \varepsilon_{4}x \\ a_6&=x^6-\varepsilon_{2}x^4 -\varepsilon_{4} \\ a_7&=x^7 \\ a_8&=x^8+\varepsilon_{8}x^6 -\varepsilon_{2} \varepsilon_{8}x^4 -\varepsilon_{4} \varepsilon_{8} \\ a_9&=x^9-\varepsilon_{2} \varepsilon_{8}x^5 -\varepsilon_{4} \varepsilon_{8}x \\ a_{10}&=x^{10}+\varepsilon_{2}x^8 -\varepsilon_{8}x^4 - \varepsilon_{4} \varepsilon_{8}x^2-\varepsilon_{2}\varepsilon_{4} \varepsilon_{8} \\ a_{11}&=x^{11}-\varepsilon_{4} \varepsilon_{8}x^3 \\ a_{12}&=x^{12}-\varepsilon_{4}x^{10} -\varepsilon_{2} \varepsilon_{4}x^8 + \varepsilon_{8}x^2+\varepsilon_{2}\varepsilon_{8} \\ a_{13}&=x^{13}-\varepsilon_{2} \varepsilon_{4}x^9 +\varepsilon_{2} \varepsilon_{8}x \\ a_{14}&=x^{14}-\varepsilon_{2}x^{12} +\varepsilon_{4}x^8 - \varepsilon_{8} \\ a_{15}&=x^{15} \\ a_{16}&=x^{16}+\varepsilon_{16}x^{14} -\varepsilon_{2} \varepsilon_{16}x^{12} +\varepsilon_{4} \varepsilon_{16}x^8 -\varepsilon_{8} \varepsilon_{16} \\ a_{17}&=x^{17}-\varepsilon_{2} \varepsilon_{16}x^{13} + \varepsilon_{4} \varepsilon_{16}x^9 - \varepsilon_{8} \varepsilon_{16}x \\ a_{18}&=x^{18}+\varepsilon_{2}x^{16} -\varepsilon_{16}x^{12} +\varepsilon _{4} \varepsilon_{16}x^{10} +\varepsilon_{2}\varepsilon_{4} \varepsilon_{16}x^8 - \varepsilon_{8} \varepsilon_{16}x^2 -\varepsilon_{2} \varepsilon_{8} \varepsilon_{16} \\ a_{19}&=x^{19}+\varepsilon_{4} \varepsilon_{16}x^{11} - \varepsilon_{8} \varepsilon_{16}x^3 \\ a_{20}&=x^{20}+\varepsilon_{4}x^{18} +\varepsilon_{2} \varepsilon_{4}x^{16} +\varepsilon_{16}x^{10} +\varepsilon_{2}\varepsilon_{16}x^8 -\varepsilon_{8} \varepsilon_{16}x^4 -\varepsilon_{4}\varepsilon_{8} \varepsilon _{16}x^2 -\varepsilon_{2}\varepsilon_{4}\varepsilon_{8} \varepsilon_{16} \\ a_{21}&=x^{21}+\varepsilon_{2} \varepsilon_{4}x^{17} +\varepsilon_{2} \varepsilon_{16}x^9 - \varepsilon_{8} \varepsilon_{16}x^5- \varepsilon_{2} \varepsilon_{4} \varepsilon_{8} \varepsilon_{16}x \end{aligned} $$