The answer is both No and Yes.
No - Because using the out of the box Home Page template, you cannot configure what you want. The standard out of the box home page template is as you see in the screenshot below. This necessarily means, the left panel components, viz., header and area below it cannot be switched with the right panel as you want.
Yes - You can create custom page templates and then use those in your application. With this approach, you can define the panels as you want and you will have full control on it. Any custom template created will be then available in the Lightning App Builder. You can find more on custom lightning page templates on the documentation here.