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I have a table that I have attempted to set up as follows (contrived example):

% Preview source code

%% LyX 2.3.2-2 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
documentclass[english]article
usepackage[T1]fontenc
usepackage[latin9]inputenc
usepackagegeometry
geometryverbose,tmargin=0.25in,bmargin=0.25cm,lmargin=0.25cm,rmargin=0.25cm
usepackagearray
usepackagemultirow
usepackagegraphicx

makeatletter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
%% Because html converters don't know tabularnewline
providecommandtabularnewline\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
usepackage[table]xcolor %for use in color links
usepackagecolortbl

makeatother

usepackagebabel
begindocument
noindent beginflushleft
rowcolors3gray!20%
begintabularccccccc
 & & multicolumn5cLorem ipsum dolor sit amettabularnewline
 & & Header & Header & Header & Header & Headertabularnewline
multirow8*cellcolorwhiterotatebox90Lorem ipsum & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
endtabularrowcolors2
parendflushleft
enddocument

Apologies in advance for this being generated code, but a commenter below asked for a complete compilable example instead of just the tabular snippet that I had had originally posted. Hopefully this will make it clearer, but I apologize in advance if it has obfuscated the problem.

Anyways, this renders as:

As seen, the sideways header that I am using on the left side of the table gets overwritten by the color of the row, interrupting the text.

What I would like to do is have that entire first column containing the left row header be white on all rows, which I tried to do by adding cellcolorwhite to the multirow cell, but as the image indicates, it only cleared the top gray strip in that cell, and didn't do anything to the rest of the first column.

How can I get the full text of the header to show up correctly?

Using multirow is the cause of your troubles.

documentclassarticle
usepackagegraphicx
usepackage[table]xcolor %for use in color links

begindocument

beginflushleft
begintabular@c
rotatebox[origin=c]90Lorem ipsumhspace2dimexprhtstrutbox+dpstrutbox
endtabular%
rowcolors3gray!20%
begintabularcccccc
 & multicolumn5cLorem ipsum dolor sit amettabularnewline
 & Header & Header & Header & Header & Headertabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
Content & ### & ### & ### & ### & ###tabularnewline
endtabularrowcolors2
endflushleft

enddocument

I have reduced the document to the minimum necessary.

Note that noindentbeginflushleft is nonsense, because it just adds unwanted vertical space (a blank line). Also parendflushleft is not necessary.

What does the strange hspace command? It adds the vertical size of the two header rows.

In the following code, I have added a cellcolorwhite command to all the cells that are in a gray row and should actually have a white background. I have also shifted the multirow command from the first row to the last row and used -8 instead of 8:

documentclass[english]article
usepackage[T1]fontenc
usepackage[latin9]inputenc
usepackagegeometry
geometryverbose,tmargin=0.25in,bmargin=0.25cm,lmargin=0.25cm,rmargin=0.25cm
usepackagearray
usepackagemultirow
usepackagegraphicx

usepackage[table]xcolor


begindocument
noindent 
beginflushleft
rowcolors3gray!20%
begintabularccccccc
 & & multicolumn5cLorem ipsum dolor sit amettabularnewline
 & & Header & Header & Header & Header & Headertabularnewline
cellcolorwhite & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
cellcolorwhite & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
cellcolorwhite & Content & ### & ### & ### & ### & ###tabularnewline
 & Content & ### & ### & ### & ### & ###tabularnewline
cellcolorwhite & Content & ### & ### & ### & ### & ###tabularnewline
multirow-8*rotatebox90Lorem ipsum & Content & ### & ### & ### & ### & ###tabularnewline
endtabular
parendflushleft
enddocument