If you define $\ue\pm$ as

$$\ue\pm= \frac{1}{\sqrt{2}}\big(\ue x \pm \mathrm{i} \ue y\big)$$

you get

$$\ue+\wedge \ue- = -\mathrm{i}\ue z.$$

If, instead, you define

$$\ue+ = -\frac{1}{\sqrt{2}}\big(\ue x + \mathrm{i} \ue y\big)$$

and

$$\ue- = \frac{1}{\sqrt{2}}\big(\ue x - \mathrm{i} \ue y\big),$$

the cross product becomes

$$\ue+\wedge \ue- = \mathrm{i}\ue z.$$

Therefore, in the former case, $(\ue+,\ue-,\mathrm{i}{\ue z})$ has the same orientation of $(\ue x,\ue y, \ue z)$, whereas in the latter the orientation is preserved by $(\ue -,\ue +, \mathrm{i}\ue z)$.

Is there any reason to prefer one orientation with respect to the other? I would say no, it's probably just a matter of tradition.

$\newcommand{\ue}[1]{\hat{\mathbf{e}}_{#1}}$ If you define $\ue\pm$ as

$$\ue\pm= \frac{1}{\sqrt{2}}\big(\ue x \pm \mathrm{i} \ue y\big)$$

you get

$$\ue+\wedge \ue- = -\mathrm{i}\ue z.$$

If, instead, you define

$$\ue+ = -\frac{1}{\sqrt{2}}\big(\ue x + \mathrm{i} \ue y\big)$$

and

$$\ue- = \frac{1}{\sqrt{2}}\big(\ue x - \mathrm{i} \ue y\big),$$

the cross product becomes

$$\ue+\wedge \ue- = \mathrm{i}\ue z.$$

Therefore, in the former case, $(\ue+,\ue-,\mathrm{i}{\ue z})$ has the same orientation of $(\ue x,\ue y, \ue z)$, whereas in the latter the orientation is preserved by $(\ue -,\ue +, \mathrm{i}\ue z)$.

Is there any reason to prefer one orientation with respect to the other? I would say no, it's probably just a matter of tradition.

If you define $\ue\pm$ as

$$\ue\pm= \frac{1}{\sqrt{2}}\big(\ue x \pm \mathrm{i} \ue y\big)$$

you get

$$\ue+\wedge \ue- = -\mathrm{i}\ue z.$$

If, instead, you define

$$\ue+ = -\frac{1}{\sqrt{2}}\big(\ue x + \mathrm{i} \ue y\big)$$

and

$$\ue- = \frac{1}{\sqrt{2}}\big(\ue x - \mathrm{i} \ue y\big),$$

the cross product becomes

$$\ue+\wedge \ue- = \mathrm{i}\ue z,$$

preserving the orientation.

If you define $\ue\pm$ as

$$\ue\pm= \frac{1}{\sqrt{2}}\big(\ue x \pm \mathrm{i} \ue y\big)$$

you get

$$\ue+\wedge \ue- = -\mathrm{i}\ue z.$$

If, instead, you define

$$\ue+ = -\frac{1}{\sqrt{2}}\big(\ue x + \mathrm{i} \ue y\big)$$

and

$$\ue- = \frac{1}{\sqrt{2}}\big(\ue x - \mathrm{i} \ue y\big),$$

the cross product becomes

$$\ue+\wedge \ue- = \mathrm{i}\ue z.$$

Therefore, in the former case, $(\ue+,\ue-,\mathrm{i}{\ue z})$ has the same orientation of $(\ue x,\ue y, \ue z)$, whereas in the latter the orientation is preserved by $(\ue -,\ue +, \mathrm{i}\ue z)$.

Is there any reason to prefer one orientation with respect to the other? I would say no, it's probably just a matter of tradition.