screenshot

I want to vertically center the diamond on the page, but preserve the position of the crown relative to the diamond.

I can't group (or select all) then center, because that centers the group using the center of the bounding-box as the midpoint.

I want to center the group, while keeping the midpoint of the group the same as the midpoint of the diamond.

I could drag the midpoint of the group to the same location as the diamond's midpoint, but the diamond's midpoint is only marked while I'm selecting just the diamond.  So I can't see it when I'm trying to center the group.

You can group crown + diamond and because crown fits inside the bounding box width of the diamond it will align to this bounding relative to the page.

That only works for horizontal centering.  I want to vertically center the diamond, while preserving the position of the crown relative to the diamond.

The buttons are confusingly labeled in Inkscape.  What's called "vertical align: middle" in CSS is called "Center on horizontal axis" in Inkscape.  In your GIF you're horizontally centering the image.  I specifically said I want to vertically center it.

This is what I did then. Why is your design not symmetrical in first place?

<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<svg
   xmlns:dc="http://purl.org/dc/elements/1.1/"
   xmlns:cc="http://creativecommons.org/ns#"
   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
   xmlns:svg="http://www.w3.org/2000/svg"
   xmlns="http://www.w3.org/2000/svg"
   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
   width="56.526794mm"
   height="59.751286mm"
   viewBox="0 0 56.526794 59.751284"
   version="1.1"
   id="svg4661"
   inkscape:version="1.0.2-2 (e86c870879, 2021-01-15)"
   sodipodi:docname="triple-diamond-director.svg">
  <title
     id="title4711">Triple Diamond Director</title>
  <defs
     id="defs4655" />
  <sodipodi:namedview
     id="base"
     pagecolor="#ffffff"
     bordercolor="#666666"
     borderopacity="1.0"
     inkscape:pageopacity="0.0"
     inkscape:pageshadow="2"
     inkscape:zoom="0.7"
     inkscape:cx="-168.38344"
     inkscape:cy="141.21355"
     inkscape:document-units="mm"
     inkscape:current-layer="layer1"
     inkscape:document-rotation="0"
     showgrid="false"
     fit-margin-top="0"
     fit-margin-left="0"
     fit-margin-right="0"
     fit-margin-bottom="0"
     inkscape:window-width="1346"
     inkscape:window-height="728"
     inkscape:window-x="4"
     inkscape:window-y="0"
     inkscape:window-maximized="0" />
  <metadata
     id="metadata4658">
    <rdf:RDF>
      <cc:Work
         rdf:about="">
        <dc:format>image/svg+xml</dc:format>
        <dc:type
           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
        <dc:title>Triple Diamond Director</dc:title>
      </cc:Work>
    </rdf:RDF>
  </metadata>
  <g
     inkscape:label="Layer 1"
     inkscape:groupmode="layer"
     id="layer1"
     transform="translate(-314.35267,-125.78605)">
    <g
       id="g3866-0-0-4"
       style="display:inline"
       transform="translate(284.89592,3.3584226)">
      <path
         style="fill:#a7c1da;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 36.189226,141.49175 h 20.24734 l -9.427765,16.32937 -17.552048,-4.70305 z"
         id="path3778-6-7-2" />
      <path
         style="fill:#8aabcd;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 30.643555,156.33518 24.337044,24.33704 -8.205494,-20.82687 -17.299844,-4.63548 z"
         id="path3780-52-1-8" />
      <path
         style="fill:#6894b5;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 57.521063,144.38181 -8.34471,13.67681 h 16.855526 z"
         id="path3782-4-5-3" />
      <path
         style="fill:#6f95be;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 49.040722,160.2358 h 16.890969 l -8.360382,21.94311 z"
         id="path3786-9-3-5" />
      <path
         style="display:inline;fill:#6e97b5;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 79.247308,141.49175 h -20.24734 l 9.427765,16.32937 17.552048,-4.70305 z"
         id="path3778-5-9-5-7" />
      <path
         style="display:inline;fill:#4b779a;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
         d="m 84.815251,156.33518 -24.337044,24.33704 8.205494,-20.82687 17.299844,-4.63548 z"
         id="path3780-5-3-2-2" />
    </g>
    <path
       style="display:inline;fill:#4a77ab;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
       d="m 321.37018,142.71971 h 42.91388 v -7.84078 c 0,0 -3.39639,4.36324 -10.67805,4.36324 -7.28165,0 -10.67756,-4.27783 -10.67756,-4.27783 0,0 -3.20659,4.26626 -10.39099,4.26626 -7.1844,0 -11.35719,-4.27821 -11.35719,-4.27821 z"
       id="path4077-4" />
    <rect
       style="display:inline;fill:#4a77ab;fill-opacity:1;stroke-width:0;stroke-miterlimit:4;stroke-dasharray:none"
       id="rect4079-2"
       width="4.1724749"
       height="4.1724749"
       x="328.89859"
       y="-141.58943"
       transform="rotate(45)" />
    <rect
       style="display:inline;fill:#4a77ab;fill-opacity:1;stroke-width:0;stroke-miterlimit:4;stroke-dasharray:none"
       id="rect4079-3-8"
       width="4.1724749"
       height="4.1724749"
       x="344.34546"
       y="-157.03629"
       transform="rotate(45)" />
    <path
       style="display:inline;fill:#4a77ab;fill-opacity:1;stroke:#000000;stroke-width:0;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
       d="m 339.92495,128.79414 2.84143,4.9215 2.89349,-5.01166 -2.91792,-2.91793 z"
       id="path4305" />
  </g>
</svg>

Checkout these screenshots.  The first shows the group vertically centered.  The second shows the diamond and crown individually centered.  

In the first screenshot, neither diamond _nor_ crown are positioned how I want.

In the second screenshot, only the _diamond_ is positioned how I want.

Does that better illustrate what I'm trying to achieve?

Maybe this:

A variation of Polygon's method:  Draw some temporary construction lines centred on the diamond and create a group.  Delete them when you're happy with the arrangement.

GirkovArpa

Does that better illustrate what I'm trying to achieve?

Unfortunately not for me. I just guess you want to maintain the white/negative space between crown and diamond. Diamond still not symmetrical. But when diamond is in the right place the crown will be cropped - Screenshot and file are not having the same page aspect ratio:

Like Dr. Frankenstein, I will animate this lifeless husk!  < cue lightning flash >

Thanks guys.  @Paddy_CAD's technique looks simplest so I think I'll try that one for now 😎  

I understand the cropping issue but I'll just resize the page.