Is there a way to link rows between sheets when sorting?

For example, I have a database in which the first several columns in a
set of rows from sheet two reference the first several columns in a
corresponding set of rows from sheet one. The remaining columns on
sheet two are used to input new data, but it is important that each row
stay connected.

Is there a way to program the database such that if the selection of
rows in sheet one is sorted that the selection of corresponding rows in
sheet two (the entire rows, not just the first several columns) will
automatically sort in the same way?

This is a different approach from what you requested, but hopefully it meets the need.

The data can be combined into one sheet instead of two, with all the columns (but only one copy of any column that now appears in both sheets). When you need a "first sheet" kind of view, hide the columns not wanted. Likewise, when you need a "second sheet" kind of view, hide the columns not wanted then.

Excel 2003 has a "custom view" feature to streamline switching views. Each custom view remembers which rows and/or columns to hide, and it also remembers any filter criteria. Using it you can switch between the two views readily. You might also find additional custom views useful; e.g., a "combined view" with nothing hidden. Excel's built-in Help describes this feature.

When the combined sheet is sorted, things don't get scrambled.

IMO, this approach is straightforward and minimizes the potential for mismatch errors.