Off the top of my head, I can think of three different methods to accomplish this effect:
The first is the one you already mentioned, which involves using 'Path → Combine' on two paths and setting the fill rule to even-odd. However, this likely won't work in a lot of situations, especially when using strokes or markers etc., and makes things a lot harder to edit.
The second involves a bunch of duplicating (or cloning) then clipping, but this adds a lot of unnecessary complexity and redundant data. It also makes things a little harder to change later as well.
The last, and the one I want to go into more detail on, uses filters. While it is probably the most advanced, may require some minor changes to the document structure and may not render correctly in a small number of viewers, the advantages of it are that it updates in real-time, works with multiple objects and even colors, and doesn't add redundant data. It's also – at least to me – the most interesting solution of the three.
First group all the objects together, that you want to add the overlay to, and move that group to a new layer. Hide that layer in the 'Layers and Objects' panel. By the way, for the best results, make sure there are no transform-attributes on any objects inside the group or on the group itself.
Next, to prepare the pattern, you could clone the repeating section a bunch, then group and clip it, but it's probably better to set up a proper SVG pattern here. To do that, select one repeating section and use 'Object → Pattern → Objects to Pattern'. This will convert your selection to a rectangle with the new pattern applied. You can now tweak the pattern with the new pattern-editor inside the 'Fill and Stroke' panel in the recent Inkscape 1.3 release:
With the 'Move patterns (in fill or stroke) along with the objects' (this button: 
) setting in the toolbar disabled, resize the rectangle to fit the whole canvas. Your canvas and document structure should look something like this:
Now you can open the 'Filters → Filter Editor...' (this panel has also been updated in the new 1.3 version), create a new filter, and apply it to the rectangle by clicking on the checkbox in the top left with it selected. Add an 'Image' effect, select the group ('g23' in my example above) and click on the 'SVG Element' button in the parameters. The 'Source of Image' input should now display the ID of the group (#g23). Next add a 'Composite' effect, connect one of the arrows to 'Source Graphic' and set the 'Operator' drop-down to 'XOR':
And we're done! Here is the result, both black and white (transparent) and in color:
